Kombinatorika u SAGE-u

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│ SageMath version 9.2, Release Date: 2020-10-24                     │
│ Using Python 3.8.6. Type "help()" for help.                        │
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%display ascii_art

Kartezijev produkt skupova

Kartezijev produkt skupova $\{1,2,3\}$ i $\{a,b\}$

cp2=cartesian_product(([1,2,3],['a','b']));cp2

The Cartesian product of ({1, 2, 3}, {'a', 'b'})

cp2.list()

[ ( 1, a ), ( 1, b ), ( 2, a ), ( 2, b ), ( 3, a ), ( 3, b ) ]

cp2.cardinality()

6

Kartezijev produkt skupova $\{4,5,1\}$, $\{-1,-2\}$ i $\{a,b,c\}$ navedenim redom

cp3=cartesian_product(([4,5,1],[-1,-2],['a','b','c']));cp3

The Cartesian product of ({4, 5, 1}, {-1, -2}, {'a', 'b', 'c'})

cp3.cardinality()

18

cp3.list()

[ ( 4, -1, a ), ( 4, -1, b ), ( 4, -1, c ), ( 4, -2, a ), ( 4, -2, b ),

 ( 4, -2, c ), ( 5, -1, a ), ( 5, -1, b ), ( 5, -1, c ), ( 5, -2, a ),

 ( 5, -2, b ), ( 5, -2, c ), ( 1, -1, a ), ( 1, -1, b ), ( 1, -1, c ),

 ( 1, -2, a ), ( 1, -2, b ), ( 1, -2, c ) ]

r-permutacije

Ispišite sve 3-permutacije skupa $\{a,b,c,d\}$.

per=Arrangements(['a','b','c','d'],3);per

Arrangements of the set ['a', 'b', 'c', 'd'] of length 3

per.list()

[ ['a', 'b', 'c'], ['a', 'b', 'd'], ['a', 'c', 'b'], ['a', 'c', 'd'],

 ['a', 'd', 'b'], ['a', 'd', 'c'], ['b', 'a', 'c'], ['b', 'a', 'd'],

 ['b', 'c', 'a'], ['b', 'c', 'd'], ['b', 'd', 'a'], ['b', 'd', 'c'],

 ['c', 'a', 'b'], ['c', 'a', 'd'], ['c', 'b', 'a'], ['c', 'b', 'd'],

 ['c', 'd', 'a'], ['c', 'd', 'b'], ['d', 'a', 'b'], ['d', 'a', 'c'],

 ['d', 'b', 'a'], ['d', 'b', 'c'], ['d', 'c', 'a'], ['d', 'c', 'b'] ]

per.cardinality()

24

r-kombinacije

Ispišite sve 3-kombinacije skupa $\{a,b,c,d\}$.

komb=Combinations(['a','b','c','d'],3);komb

Combinations of ['a', 'b', 'c', 'd'] of length 3

komb.list()

[ [ a, b, c ], [ a, b, d ], [ a, c, d ], [ b, c, d ] ]

komb.cardinality()

4

r-permutacije s ponavljanjem

Ispišite sve 3-permutacije s ponavljanjem skupa $\{a,b,c,d\}$.

perpon=Tuples(['a','b','c','d'],3);perpon

Tuples of ('a', 'b', 'c', 'd') of length 3

perpon.list()

[ [ a, a, a ], [ b, a, a ], [ c, a, a ], [ d, a, a ], [ a, b, a ], [ b, b, a ],

 [ c, b, a ], [ d, b, a ], [ a, c, a ], [ b, c, a ], [ c, c, a ], [ d, c, a ],

 [ a, d, a ], [ b, d, a ], [ c, d, a ], [ d, d, a ], [ a, a, b ], [ b, a, b ],

 [ c, a, b ], [ d, a, b ], [ a, b, b ], [ b, b, b ], [ c, b, b ], [ d, b, b ],

 [ a, c, b ], [ b, c, b ], [ c, c, b ], [ d, c, b ], [ a, d, b ], [ b, d, b ],

 [ c, d, b ], [ d, d, b ], [ a, a, c ], [ b, a, c ], [ c, a, c ], [ d, a, c ],

 [ a, b, c ], [ b, b, c ], [ c, b, c ], [ d, b, c ], [ a, c, c ], [ b, c, c ],

 [ c, c, c ], [ d, c, c ], [ a, d, c ], [ b, d, c ], [ c, d, c ], [ d, d, c ],

 [ a, a, d ], [ b, a, d ], [ c, a, d ], [ d, a, d ], [ a, b, d ], [ b, b, d ],

 [ c, b, d ], [ d, b, d ], [ a, c, d ], [ b, c, d ], [ c, c, d ], [ d, c, d ],

 [ a, d, d ], [ b, d, d ], [ c, d, d ], [ d, d, d ] ]

perpon.cardinality()

64

r-kombinacije s ponavljanjem

Ispišite sve 3-kombinacije s ponavljanjem skupa $\{a,b,c,d\}$.

kombpon=UnorderedTuples(['a','b','c','d'],3);kombpon

Unordered tuples of ('a', 'b', 'c', 'd') of length 3

kombpon.list()

[ [ a, a, a ], [ a, a, b ], [ a, a, c ], [ a, a, d ], [ a, b, b ], [ a, b, c ],

 [ a, b, d ], [ a, c, c ], [ a, c, d ], [ a, d, d ], [ b, b, b ], [ b, b, c ],

 [ b, b, d ], [ b, c, c ], [ b, c, d ], [ b, d, d ], [ c, c, c ], [ c, c, d ],

 [ c, d, d ], [ d, d, d ] ]

kombpon.cardinality()

20

Permutacije i cikličke permutacije skupa

Ispišite sve permutacije skupa $\{a,b,c,d\}$.

p=Permutations(['a','b','c','d']);p

Permutations of the set ['a', 'b', 'c', 'd']

p.cardinality()

24

p.list()

[ ['a', 'b', 'c', 'd'], ['a', 'b', 'd', 'c'], ['a', 'c', 'b', 'd'],

 ['a', 'c', 'd', 'b'], ['a', 'd', 'b', 'c'], ['a', 'd', 'c', 'b'],

 ['b', 'a', 'c', 'd'], ['b', 'a', 'd', 'c'], ['b', 'c', 'a', 'd'],

 ['b', 'c', 'd', 'a'], ['b', 'd', 'a', 'c'], ['b', 'd', 'c', 'a'],

 ['c', 'a', 'b', 'd'], ['c', 'a', 'd', 'b'], ['c', 'b', 'a', 'd'],

 ['c', 'b', 'd', 'a'], ['c', 'd', 'a', 'b'], ['c', 'd', 'b', 'a'],

 ['d', 'a', 'b', 'c'], ['d', 'a', 'c', 'b'], ['d', 'b', 'a', 'c'],

 ['d', 'b', 'c', 'a'], ['d', 'c', 'a', 'b'], ['d', 'c', 'b', 'a'] ]

Ispišite sve cikličke permutacije skupa $\{a,b,c,d\}$.

cp=CyclicPermutations(['a','b','c','d']);cp

Cyclic permutations of ['a', 'b', 'c', 'd']

len(cp.list())

6

cp.list()

[ [ a, b, c, d ], [ a, b, d, c ], [ a, c, b, d ], [ a, c, d, b ],

 [ a, d, b, c ], [ a, d, c, b ] ]

Deranžmani

Napišite sve deranžmane skupa $\{a,b,c,d\}$.

der=Derangements(['a','b','c','d']);der

Derangements of the set ['a', 'b', 'c', 'd']

len(der)

9

der.list()

[ ['b', 'c', 'd', 'a'], ['d', 'c', 'a', 'b'], ['b', 'd', 'a', 'c'],

 ['c', 'd', 'b', 'a'], ['c', 'a', 'd', 'b'], ['d', 'a', 'b', 'c'],

 ['d', 'c', 'b', 'a'], ['c', 'd', 'a', 'b'], ['b', 'a', 'd', 'c'] ]

Nekoliko konkretnih zadataka

1. zadatak

Koliko ima četveroznamenkastih prirodnih brojeva s različitim znamenkama?

Rješenje

brojevi=list(filter(lambda x: x[0]!=0 and len(set(x))==4, Tuples(range(10),4)))

len(brojevi)

4536

prvih 200 takvih brojeva

brojevi=sorted(map(lambda x: int(''.join(map(str,x))), brojevi))

brojevi[0:200]

[ 1023, 1024, 1025, 1026, 1027, 1028, 1029, 1032, 1034, 1035, 1036, 1037, 1038,

 1039, 1042, 1043, 1045, 1046, 1047, 1048, 1049, 1052, 1053, 1054, 1056, 1057,

 1058, 1059, 1062, 1063, 1064, 1065, 1067, 1068, 1069, 1072, 1073, 1074, 1075,

 1076, 1078, 1079, 1082, 1083, 1084, 1085, 1086, 1087, 1089, 1092, 1093, 1094,

 1095, 1096, 1097, 1098, 1203, 1204, 1205, 1206, 1207, 1208, 1209, 1230, 1234,

 1235, 1236, 1237, 1238, 1239, 1240, 1243, 1245, 1246, 1247, 1248, 1249, 1250,

 1253, 1254, 1256, 1257, 1258, 1259, 1260, 1263, 1264, 1265, 1267, 1268, 1269,

 1270, 1273, 1274, 1275, 1276, 1278, 1279, 1280, 1283, 1284, 1285, 1286, 1287,

 1289, 1290, 1293, 1294, 1295, 1296, 1297, 1298, 1302, 1304, 1305, 1306, 1307,

 1308, 1309, 1320, 1324, 1325, 1326, 1327, 1328, 1329, 1340, 1342, 1345, 1346,

 1347, 1348, 1349, 1350, 1352, 1354, 1356, 1357, 1358, 1359, 1360, 1362, 1364,

 1365, 1367, 1368, 1369, 1370, 1372, 1374, 1375, 1376, 1378, 1379, 1380, 1382,

 1384, 1385, 1386, 1387, 1389, 1390, 1392, 1394, 1395, 1396, 1397, 1398, 1402,

 1403, 1405, 1406, 1407, 1408, 1409, 1420, 1423, 1425, 1426, 1427, 1428, 1429,

 1430, 1432, 1435, 1436, 1437, 1438, 1439, 1450, 1452, 1453, 1456, 1457, 1458,

 1459, 1460, 1462, 1463, 1465 ]

2. zadatak

Koliko ima prirodnih brojeva strogo manjih od 10000 s različitim znamenkama?

Rješenje

br1=list(Tuples(range(1,10),1))
br2=list(filter(lambda x: x[0]!=0 and len(set(x))==2, Tuples(range(10),2)))
br3=list(filter(lambda x: x[0]!=0 and len(set(x))==3, Tuples(range(10),3)))
br4=list(filter(lambda x: x[0]!=0 and len(set(x))==4, Tuples(range(10),4)))
brojevi2=br1 + br2 + br3 + br4

len(brojevi2)

5274

prvih 400 takvih brojeva

brojevi2=sorted(map(lambda x: int(''.join(map(str,x))), brojevi2))

brojevi2[0:400]

[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24,

 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46,

 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68,

 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90,

 91, 92, 93, 94, 95, 96, 97, 98, 102, 103, 104, 105, 106, 107, 108, 109, 120,

 123, 124, 125, 126, 127, 128, 129, 130, 132, 134, 135, 136, 137, 138, 139, 140,

 142, 143, 145, 146, 147, 148, 149, 150, 152, 153, 154, 156, 157, 158, 159, 160,

 162, 163, 164, 165, 167, 168, 169, 170, 172, 173, 174, 175, 176, 178, 179, 180,

 182, 183, 184, 185, 186, 187, 189, 190, 192, 193, 194, 195, 196, 197, 198, 201,

 203, 204, 205, 206, 207, 208, 209, 210, 213, 214, 215, 216, 217, 218, 219, 230,

 231, 234, 235, 236, 237, 238, 239, 240, 241, 243, 245, 246, 247, 248, 249, 250,

 251, 253, 254, 256, 257, 258, 259, 260, 261, 263, 264, 265, 267, 268, 269, 270,

 271, 273, 274, 275, 276, 278, 279, 280, 281, 283, 284, 285, 286, 287, 289, 290,

 291, 293, 294, 295, 296, 297, 298, 301, 302, 304, 305, 306, 307, 308, 309, 310,

 312, 314, 315, 316, 317, 318, 319, 320, 321, 324, 325, 326, 327, 328, 329, 340,

 341, 342, 345, 346, 347, 348, 349, 350, 351, 352, 354, 356, 357, 358, 359, 360,

 361, 362, 364, 365, 367, 368, 369, 370, 371, 372, 374, 375, 376, 378, 379, 380,

 381, 382, 384, 385, 386, 387, 389, 390, 391, 392, 394, 395, 396, 397, 398, 401,

 402, 403, 405, 406, 407, 408, 409, 410, 412, 413, 415, 416, 417, 418, 419, 420,

 421, 423, 425, 426, 427, 428, 429, 430, 431, 432, 435, 436, 437, 438, 439, 450,

 451, 452, 453, 456, 457, 458, 459, 460, 461, 462, 463, 465, 467, 468, 469, 470,

 471, 472, 473, 475, 476, 478, 479, 480, 481, 482, 483, 485, 486, 487, 489, 490,

 491, 492, 493, 495, 496, 497, 498, 501, 502, 503, 504, 506, 507, 508, 509, 510,

 512, 513, 514, 516, 517, 518, 519, 520, 521, 523, 524, 526, 527 ]

3. zadatak

Koliko ima neparnih četveroznamenkastih prirodnih brojeva s različitim znamenkama?

Rješenje

neparni4=list(filter(lambda x: x[0]!=0 and len(set(x))==4 and (x[-1]%2==1), Tuples(range(10),4)))

len(neparni4)

2240

prvih 200 takvih brojeva

neparni4=sorted(map(lambda x: int(''.join(map(str,x))), neparni4))

neparni4[0:200]

[ 1023, 1025, 1027, 1029, 1035, 1037, 1039, 1043, 1045, 1047, 1049, 1053, 1057,

 1059, 1063, 1065, 1067, 1069, 1073, 1075, 1079, 1083, 1085, 1087, 1089, 1093,

 1095, 1097, 1203, 1205, 1207, 1209, 1235, 1237, 1239, 1243, 1245, 1247, 1249,

 1253, 1257, 1259, 1263, 1265, 1267, 1269, 1273, 1275, 1279, 1283, 1285, 1287,

 1289, 1293, 1295, 1297, 1305, 1307, 1309, 1325, 1327, 1329, 1345, 1347, 1349,

 1357, 1359, 1365, 1367, 1369, 1375, 1379, 1385, 1387, 1389, 1395, 1397, 1403,

 1405, 1407, 1409, 1423, 1425, 1427, 1429, 1435, 1437, 1439, 1453, 1457, 1459,

 1463, 1465, 1467, 1469, 1473, 1475, 1479, 1483, 1485, 1487, 1489, 1493, 1495,

 1497, 1503, 1507, 1509, 1523, 1527, 1529, 1537, 1539, 1543, 1547, 1549, 1563,

 1567, 1569, 1573, 1579, 1583, 1587, 1589, 1593, 1597, 1603, 1605, 1607, 1609,

 1623, 1625, 1627, 1629, 1635, 1637, 1639, 1643, 1645, 1647, 1649, 1653, 1657,

 1659, 1673, 1675, 1679, 1683, 1685, 1687, 1689, 1693, 1695, 1697, 1703, 1705,

 1709, 1723, 1725, 1729, 1735, 1739, 1743, 1745, 1749, 1753, 1759, 1763, 1765,

 1769, 1783, 1785, 1789, 1793, 1795, 1803, 1805, 1807, 1809, 1823, 1825, 1827,

 1829, 1835, 1837, 1839, 1843, 1845, 1847, 1849, 1853, 1857, 1859, 1863, 1865,

 1867, 1869, 1873, 1875, 1879 ]

4. zadatak

Koliko se različitih riječi (smislenih i besmislenih) može dobiti permutiranjem riječi MINIMUM ?

Rješenje

Permutations('MINIMUM').cardinality()

420

rijeci=Permutations('MINIMUM').list()
list(map(lambda x: ''.join(x),rijeci))

[ MMMIINU, MMMIIUN, MMMINIU, MMMINUI, MMMIUIN, MMMIUNI, MMMNIIU, MMMNIUI,

 MMMNUII, MMMUIIN, MMMUINI, MMMUNII, MMIMINU, MMIMIUN, MMIMNIU, MMIMNUI,

 MMIMUIN, MMIMUNI, MMIIMNU, MMIIMUN, MMIINMU, MMIINUM, MMIIUMN, MMIIUNM,

 MMINMIU, MMINMUI, MMINIMU, MMINIUM, MMINUMI, MMINUIM, MMIUMIN, MMIUMNI,

 MMIUIMN, MMIUINM, MMIUNMI, MMIUNIM, MMNMIIU, MMNMIUI, MMNMUII, MMNIMIU,

 MMNIMUI, MMNIIMU, MMNIIUM, MMNIUMI, MMNIUIM, MMNUMII, MMNUIMI, MMNUIIM,

 MMUMIIN, MMUMINI, MMUMNII, MMUIMIN, MMUIMNI, MMUIIMN, MMUIINM, MMUINMI,

 MMUINIM, MMUNMII, MMUNIMI, MMUNIIM, MIMMINU, MIMMIUN, MIMMNIU, MIMMNUI,

 MIMMUIN, MIMMUNI, MIMIMNU, MIMIMUN, MIMINMU, MIMINUM, MIMIUMN, MIMIUNM,

 MIMNMIU, MIMNMUI, MIMNIMU, MIMNIUM, MIMNUMI, MIMNUIM, MIMUMIN, MIMUMNI,

 MIMUIMN, MIMUINM, MIMUNMI, MIMUNIM, MIIMMNU, MIIMMUN, MIIMNMU, MIIMNUM,

 MIIMUMN, MIIMUNM, MIINMMU, MIINMUM, MIINUMM, MIIUMMN, MIIUMNM, MIIUNMM,

 MINMMIU, MINMMUI, MINMIMU, MINMIUM, MINMUMI, MINMUIM, MINIMMU, MINIMUM,

 MINIUMM, MINUMMI, MINUMIM, MINUIMM, MIUMMIN, MIUMMNI, MIUMIMN, MIUMINM,

 MIUMNMI, MIUMNIM, MIUIMMN, MIUIMNM, MIUINMM, MIUNMMI, MIUNMIM, MIUNIMM,

 MNMMIIU, MNMMIUI, MNMMUII, MNMIMIU, MNMIMUI, MNMIIMU, MNMIIUM, MNMIUMI,

 MNMIUIM, MNMUMII, MNMUIMI, MNMUIIM, MNIMMIU, MNIMMUI, MNIMIMU, MNIMIUM,

 MNIMUMI, MNIMUIM, MNIIMMU, MNIIMUM, MNIIUMM, MNIUMMI, MNIUMIM, MNIUIMM,

 MNUMMII, MNUMIMI, MNUMIIM, MNUIMMI, MNUIMIM, MNUIIMM, MUMMIIN, MUMMINI,

 MUMMNII, MUMIMIN, MUMIMNI, MUMIIMN, MUMIINM, MUMINMI, MUMINIM, MUMNMII,

 MUMNIMI, MUMNIIM, MUIMMIN, MUIMMNI, MUIMIMN, MUIMINM, MUIMNMI, MUIMNIM,

 MUIIMMN, MUIIMNM, MUIINMM, MUINMMI, MUINMIM, MUINIMM, MUNMMII, MUNMIMI,

 MUNMIIM, MUNIMMI, MUNIMIM, MUNIIMM, IMMMINU, IMMMIUN, IMMMNIU, IMMMNUI,

 IMMMUIN, IMMMUNI, IMMIMNU, IMMIMUN, IMMINMU, IMMINUM, IMMIUMN, IMMIUNM,

 IMMNMIU, IMMNMUI, IMMNIMU, IMMNIUM, IMMNUMI, IMMNUIM, IMMUMIN, IMMUMNI,

 IMMUIMN, IMMUINM, IMMUNMI, IMMUNIM, IMIMMNU, IMIMMUN, IMIMNMU, IMIMNUM,

 IMIMUMN, IMIMUNM, IMINMMU, IMINMUM, IMINUMM, IMIUMMN, IMIUMNM, IMIUNMM,

 IMNMMIU, IMNMMUI, IMNMIMU, IMNMIUM, IMNMUMI, IMNMUIM, IMNIMMU, IMNIMUM,

 IMNIUMM, IMNUMMI, IMNUMIM, IMNUIMM, IMUMMIN, IMUMMNI, IMUMIMN, IMUMINM,

 IMUMNMI, IMUMNIM, IMUIMMN, IMUIMNM, IMUINMM, IMUNMMI, IMUNMIM, IMUNIMM,

 IIMMMNU, IIMMMUN, IIMMNMU, IIMMNUM, IIMMUMN, IIMMUNM, IIMNMMU, IIMNMUM,

 IIMNUMM, IIMUMMN, IIMUMNM, IIMUNMM, IINMMMU, IINMMUM, IINMUMM, IINUMMM,

 IIUMMMN, IIUMMNM, IIUMNMM, IIUNMMM, INMMMIU, INMMMUI, INMMIMU, INMMIUM,

 INMMUMI, INMMUIM, INMIMMU, INMIMUM, INMIUMM, INMUMMI, INMUMIM, INMUIMM,

 INIMMMU, INIMMUM, INIMUMM, INIUMMM, INUMMMI, INUMMIM, INUMIMM, INUIMMM,

 IUMMMIN, IUMMMNI, IUMMIMN, IUMMINM, IUMMNMI, IUMMNIM, IUMIMMN, IUMIMNM,

 IUMINMM, IUMNMMI, IUMNMIM, IUMNIMM, IUIMMMN, IUIMMNM, IUIMNMM, IUINMMM,

 IUNMMMI, IUNMMIM, IUNMIMM, IUNIMMM, NMMMIIU, NMMMIUI, NMMMUII, NMMIMIU,

 NMMIMUI, NMMIIMU, NMMIIUM, NMMIUMI, NMMIUIM, NMMUMII, NMMUIMI, NMMUIIM,

 NMIMMIU, NMIMMUI, NMIMIMU, NMIMIUM, NMIMUMI, NMIMUIM, NMIIMMU, NMIIMUM,

 NMIIUMM, NMIUMMI, NMIUMIM, NMIUIMM, NMUMMII, NMUMIMI, NMUMIIM, NMUIMMI,

 NMUIMIM, NMUIIMM, NIMMMIU, NIMMMUI, NIMMIMU, NIMMIUM, NIMMUMI, NIMMUIM,

 NIMIMMU, NIMIMUM, NIMIUMM, NIMUMMI, NIMUMIM, NIMUIMM, NIIMMMU, NIIMMUM,

 NIIMUMM, NIIUMMM, NIUMMMI, NIUMMIM, NIUMIMM, NIUIMMM, NUMMMII, NUMMIMI,

 NUMMIIM, NUMIMMI, NUMIMIM, NUMIIMM, NUIMMMI, NUIMMIM, NUIMIMM, NUIIMMM,

 UMMMIIN, UMMMINI, UMMMNII, UMMIMIN, UMMIMNI, UMMIIMN, UMMIINM, UMMINMI,

 UMMINIM, UMMNMII, UMMNIMI, UMMNIIM, UMIMMIN, UMIMMNI, UMIMIMN, UMIMINM,

 UMIMNMI, UMIMNIM, UMIIMMN, UMIIMNM, UMIINMM, UMINMMI, UMINMIM, UMINIMM,

 UMNMMII, UMNMIMI, UMNMIIM, UMNIMMI, UMNIMIM, UMNIIMM, UIMMMIN, UIMMMNI,

 UIMMIMN, UIMMINM, UIMMNMI, UIMMNIM, UIMIMMN, UIMIMNM, UIMINMM, UIMNMMI,

 UIMNMIM, UIMNIMM, UIIMMMN, UIIMMNM, UIIMNMM, UIINMMM, UINMMMI, UINMMIM,

 UINMIMM, UINIMMM, UNMMMII, UNMMIMI, UNMMIIM, UNMIMMI, UNMIMIM, UNMIIMM,

 UNIMMMI, UNIMMIM, UNIMIMM, UNIIMMM ]

5. zadatak

Koliko permutacija skupa $\{1,2,3,4,5,6\}$

1. počinje brojem $4$,
2. ima na prva dva mjesta broj $1, 2$ ili $3$,
3. završava neparnim brojem,
4. završava brojem $21$,
5. ne završava brojem $234$?

Rješenje

permS=Permutations([1,2,3,4,5,6]).list()

a) dio

len(list(filter(lambda x: x[0]==4,permS)))

120

list(filter(lambda x: x[0]==4, permS))

[ [4, 1, 2, 3, 5, 6], [4, 1, 2, 3, 6, 5], [4, 1, 2, 5, 3, 6],

 [4, 1, 2, 5, 6, 3], [4, 1, 2, 6, 3, 5], [4, 1, 2, 6, 5, 3], [4, 1, 3, 2, 5, 6],

 [4, 1, 3, 2, 6, 5], [4, 1, 3, 5, 2, 6], [4, 1, 3, 5, 6, 2], [4, 1, 3, 6, 2, 5],

 [4, 1, 3, 6, 5, 2], [4, 1, 5, 2, 3, 6], [4, 1, 5, 2, 6, 3], [4, 1, 5, 3, 2, 6],

 [4, 1, 5, 3, 6, 2], [4, 1, 5, 6, 2, 3], [4, 1, 5, 6, 3, 2], [4, 1, 6, 2, 3, 5],

 [4, 1, 6, 2, 5, 3], [4, 1, 6, 3, 2, 5], [4, 1, 6, 3, 5, 2], [4, 1, 6, 5, 2, 3],

 [4, 1, 6, 5, 3, 2], [4, 2, 1, 3, 5, 6], [4, 2, 1, 3, 6, 5], [4, 2, 1, 5, 3, 6],

 [4, 2, 1, 5, 6, 3], [4, 2, 1, 6, 3, 5], [4, 2, 1, 6, 5, 3], [4, 2, 3, 1, 5, 6],

 [4, 2, 3, 1, 6, 5], [4, 2, 3, 5, 1, 6], [4, 2, 3, 5, 6, 1], [4, 2, 3, 6, 1, 5],

 [4, 2, 3, 6, 5, 1], [4, 2, 5, 1, 3, 6], [4, 2, 5, 1, 6, 3], [4, 2, 5, 3, 1, 6],

 [4, 2, 5, 3, 6, 1], [4, 2, 5, 6, 1, 3], [4, 2, 5, 6, 3, 1], [4, 2, 6, 1, 3, 5],

 [4, 2, 6, 1, 5, 3], [4, 2, 6, 3, 1, 5], [4, 2, 6, 3, 5, 1], [4, 2, 6, 5, 1, 3],

 [4, 2, 6, 5, 3, 1], [4, 3, 1, 2, 5, 6], [4, 3, 1, 2, 6, 5], [4, 3, 1, 5, 2, 6],

 [4, 3, 1, 5, 6, 2], [4, 3, 1, 6, 2, 5], [4, 3, 1, 6, 5, 2], [4, 3, 2, 1, 5, 6],

 [4, 3, 2, 1, 6, 5], [4, 3, 2, 5, 1, 6], [4, 3, 2, 5, 6, 1], [4, 3, 2, 6, 1, 5],

 [4, 3, 2, 6, 5, 1], [4, 3, 5, 1, 2, 6], [4, 3, 5, 1, 6, 2], [4, 3, 5, 2, 1, 6],

 [4, 3, 5, 2, 6, 1], [4, 3, 5, 6, 1, 2], [4, 3, 5, 6, 2, 1], [4, 3, 6, 1, 2, 5],

 [4, 3, 6, 1, 5, 2], [4, 3, 6, 2, 1, 5], [4, 3, 6, 2, 5, 1], [4, 3, 6, 5, 1, 2],

 [4, 3, 6, 5, 2, 1], [4, 5, 1, 2, 3, 6], [4, 5, 1, 2, 6, 3], [4, 5, 1, 3, 2, 6],

 [4, 5, 1, 3, 6, 2], [4, 5, 1, 6, 2, 3], [4, 5, 1, 6, 3, 2], [4, 5, 2, 1, 3, 6],

 [4, 5, 2, 1, 6, 3], [4, 5, 2, 3, 1, 6], [4, 5, 2, 3, 6, 1], [4, 5, 2, 6, 1, 3],

 [4, 5, 2, 6, 3, 1], [4, 5, 3, 1, 2, 6], [4, 5, 3, 1, 6, 2], [4, 5, 3, 2, 1, 6],

 [4, 5, 3, 2, 6, 1], [4, 5, 3, 6, 1, 2], [4, 5, 3, 6, 2, 1], [4, 5, 6, 1, 2, 3],

 [4, 5, 6, 1, 3, 2], [4, 5, 6, 2, 1, 3], [4, 5, 6, 2, 3, 1], [4, 5, 6, 3, 1, 2],

 [4, 5, 6, 3, 2, 1], [4, 6, 1, 2, 3, 5], [4, 6, 1, 2, 5, 3], [4, 6, 1, 3, 2, 5],

 [4, 6, 1, 3, 5, 2], [4, 6, 1, 5, 2, 3], [4, 6, 1, 5, 3, 2], [4, 6, 2, 1, 3, 5],

 [4, 6, 2, 1, 5, 3], [4, 6, 2, 3, 1, 5], [4, 6, 2, 3, 5, 1], [4, 6, 2, 5, 1, 3],

 [4, 6, 2, 5, 3, 1], [4, 6, 3, 1, 2, 5], [4, 6, 3, 1, 5, 2], [4, 6, 3, 2, 1, 5],

 [4, 6, 3, 2, 5, 1], [4, 6, 3, 5, 1, 2], [4, 6, 3, 5, 2, 1], [4, 6, 5, 1, 2, 3],

 [4, 6, 5, 1, 3, 2], [4, 6, 5, 2, 1, 3], [4, 6, 5, 2, 3, 1], [4, 6, 5, 3, 1, 2],

 [4, 6, 5, 3, 2, 1] ]

b) dio

len(list(filter(lambda x: set(x[0:2]).issubset(set([1,2,3])), permS)))

144

list(filter(lambda x: set(x[0:2]).issubset(set([1,2,3])),permS))

[ [1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 6, 5], [1, 2, 3, 5, 4, 6],

 [1, 2, 3, 5, 6, 4], [1, 2, 3, 6, 4, 5], [1, 2, 3, 6, 5, 4], [1, 2, 4, 3, 5, 6],

 [1, 2, 4, 3, 6, 5], [1, 2, 4, 5, 3, 6], [1, 2, 4, 5, 6, 3], [1, 2, 4, 6, 3, 5],

 [1, 2, 4, 6, 5, 3], [1, 2, 5, 3, 4, 6], [1, 2, 5, 3, 6, 4], [1, 2, 5, 4, 3, 6],

 [1, 2, 5, 4, 6, 3], [1, 2, 5, 6, 3, 4], [1, 2, 5, 6, 4, 3], [1, 2, 6, 3, 4, 5],

 [1, 2, 6, 3, 5, 4], [1, 2, 6, 4, 3, 5], [1, 2, 6, 4, 5, 3], [1, 2, 6, 5, 3, 4],

 [1, 2, 6, 5, 4, 3], [1, 3, 2, 4, 5, 6], [1, 3, 2, 4, 6, 5], [1, 3, 2, 5, 4, 6],

 [1, 3, 2, 5, 6, 4], [1, 3, 2, 6, 4, 5], [1, 3, 2, 6, 5, 4], [1, 3, 4, 2, 5, 6],

 [1, 3, 4, 2, 6, 5], [1, 3, 4, 5, 2, 6], [1, 3, 4, 5, 6, 2], [1, 3, 4, 6, 2, 5],

 [1, 3, 4, 6, 5, 2], [1, 3, 5, 2, 4, 6], [1, 3, 5, 2, 6, 4], [1, 3, 5, 4, 2, 6],

 [1, 3, 5, 4, 6, 2], [1, 3, 5, 6, 2, 4], [1, 3, 5, 6, 4, 2], [1, 3, 6, 2, 4, 5],

 [1, 3, 6, 2, 5, 4], [1, 3, 6, 4, 2, 5], [1, 3, 6, 4, 5, 2], [1, 3, 6, 5, 2, 4],

 [1, 3, 6, 5, 4, 2], [2, 1, 3, 4, 5, 6], [2, 1, 3, 4, 6, 5], [2, 1, 3, 5, 4, 6],

 [2, 1, 3, 5, 6, 4], [2, 1, 3, 6, 4, 5], [2, 1, 3, 6, 5, 4], [2, 1, 4, 3, 5, 6],

 [2, 1, 4, 3, 6, 5], [2, 1, 4, 5, 3, 6], [2, 1, 4, 5, 6, 3], [2, 1, 4, 6, 3, 5],

 [2, 1, 4, 6, 5, 3], [2, 1, 5, 3, 4, 6], [2, 1, 5, 3, 6, 4], [2, 1, 5, 4, 3, 6],

 [2, 1, 5, 4, 6, 3], [2, 1, 5, 6, 3, 4], [2, 1, 5, 6, 4, 3], [2, 1, 6, 3, 4, 5],

 [2, 1, 6, 3, 5, 4], [2, 1, 6, 4, 3, 5], [2, 1, 6, 4, 5, 3], [2, 1, 6, 5, 3, 4],

 [2, 1, 6, 5, 4, 3], [2, 3, 1, 4, 5, 6], [2, 3, 1, 4, 6, 5], [2, 3, 1, 5, 4, 6],

 [2, 3, 1, 5, 6, 4], [2, 3, 1, 6, 4, 5], [2, 3, 1, 6, 5, 4], [2, 3, 4, 1, 5, 6],

 [2, 3, 4, 1, 6, 5], [2, 3, 4, 5, 1, 6], [2, 3, 4, 5, 6, 1], [2, 3, 4, 6, 1, 5],

 [2, 3, 4, 6, 5, 1], [2, 3, 5, 1, 4, 6], [2, 3, 5, 1, 6, 4], [2, 3, 5, 4, 1, 6],

 [2, 3, 5, 4, 6, 1], [2, 3, 5, 6, 1, 4], [2, 3, 5, 6, 4, 1], [2, 3, 6, 1, 4, 5],

 [2, 3, 6, 1, 5, 4], [2, 3, 6, 4, 1, 5], [2, 3, 6, 4, 5, 1], [2, 3, 6, 5, 1, 4],

 [2, 3, 6, 5, 4, 1], [3, 1, 2, 4, 5, 6], [3, 1, 2, 4, 6, 5], [3, 1, 2, 5, 4, 6],

 [3, 1, 2, 5, 6, 4], [3, 1, 2, 6, 4, 5], [3, 1, 2, 6, 5, 4], [3, 1, 4, 2, 5, 6],

 [3, 1, 4, 2, 6, 5], [3, 1, 4, 5, 2, 6], [3, 1, 4, 5, 6, 2], [3, 1, 4, 6, 2, 5],

 [3, 1, 4, 6, 5, 2], [3, 1, 5, 2, 4, 6], [3, 1, 5, 2, 6, 4], [3, 1, 5, 4, 2, 6],

 [3, 1, 5, 4, 6, 2], [3, 1, 5, 6, 2, 4], [3, 1, 5, 6, 4, 2], [3, 1, 6, 2, 4, 5],

 [3, 1, 6, 2, 5, 4], [3, 1, 6, 4, 2, 5], [3, 1, 6, 4, 5, 2], [3, 1, 6, 5, 2, 4],

 [3, 1, 6, 5, 4, 2], [3, 2, 1, 4, 5, 6], [3, 2, 1, 4, 6, 5], [3, 2, 1, 5, 4, 6],

 [3, 2, 1, 5, 6, 4], [3, 2, 1, 6, 4, 5], [3, 2, 1, 6, 5, 4], [3, 2, 4, 1, 5, 6],

 [3, 2, 4, 1, 6, 5], [3, 2, 4, 5, 1, 6], [3, 2, 4, 5, 6, 1], [3, 2, 4, 6, 1, 5],

 [3, 2, 4, 6, 5, 1], [3, 2, 5, 1, 4, 6], [3, 2, 5, 1, 6, 4], [3, 2, 5, 4, 1, 6],

 [3, 2, 5, 4, 6, 1], [3, 2, 5, 6, 1, 4], [3, 2, 5, 6, 4, 1], [3, 2, 6, 1, 4, 5],

 [3, 2, 6, 1, 5, 4], [3, 2, 6, 4, 1, 5], [3, 2, 6, 4, 5, 1], [3, 2, 6, 5, 1, 4],

 [3, 2, 6, 5, 4, 1] ]

c) dio

len(list(filter(lambda x: x[-1]%2==1,permS)))

360

list(filter(lambda x: x[-1]%2==1,permS))

[ [1, 2, 3, 4, 6, 5], [1, 2, 3, 6, 4, 5], [1, 2, 4, 3, 6, 5],

 [1, 2, 4, 5, 6, 3], [1, 2, 4, 6, 3, 5], [1, 2, 4, 6, 5, 3], [1, 2, 5, 4, 6, 3],

 [1, 2, 5, 6, 4, 3], [1, 2, 6, 3, 4, 5], [1, 2, 6, 4, 3, 5], [1, 2, 6, 4, 5, 3],

 [1, 2, 6, 5, 4, 3], [1, 3, 2, 4, 6, 5], [1, 3, 2, 6, 4, 5], [1, 3, 4, 2, 6, 5],

 [1, 3, 4, 6, 2, 5], [1, 3, 6, 2, 4, 5], [1, 3, 6, 4, 2, 5], [1, 4, 2, 3, 6, 5],

 [1, 4, 2, 5, 6, 3], [1, 4, 2, 6, 3, 5], [1, 4, 2, 6, 5, 3], [1, 4, 3, 2, 6, 5],

 [1, 4, 3, 6, 2, 5], [1, 4, 5, 2, 6, 3], [1, 4, 5, 6, 2, 3], [1, 4, 6, 2, 3, 5],

 [1, 4, 6, 2, 5, 3], [1, 4, 6, 3, 2, 5], [1, 4, 6, 5, 2, 3], [1, 5, 2, 4, 6, 3],

 [1, 5, 2, 6, 4, 3], [1, 5, 4, 2, 6, 3], [1, 5, 4, 6, 2, 3], [1, 5, 6, 2, 4, 3],

 [1, 5, 6, 4, 2, 3], [1, 6, 2, 3, 4, 5], [1, 6, 2, 4, 3, 5], [1, 6, 2, 4, 5, 3],

 [1, 6, 2, 5, 4, 3], [1, 6, 3, 2, 4, 5], [1, 6, 3, 4, 2, 5], [1, 6, 4, 2, 3, 5],

 [1, 6, 4, 2, 5, 3], [1, 6, 4, 3, 2, 5], [1, 6, 4, 5, 2, 3], [1, 6, 5, 2, 4, 3],

 [1, 6, 5, 4, 2, 3], [2, 1, 3, 4, 6, 5], [2, 1, 3, 6, 4, 5], [2, 1, 4, 3, 6, 5],

 [2, 1, 4, 5, 6, 3], [2, 1, 4, 6, 3, 5], [2, 1, 4, 6, 5, 3], [2, 1, 5, 4, 6, 3],

 [2, 1, 5, 6, 4, 3], [2, 1, 6, 3, 4, 5], [2, 1, 6, 4, 3, 5], [2, 1, 6, 4, 5, 3],

 [2, 1, 6, 5, 4, 3], [2, 3, 1, 4, 6, 5], [2, 3, 1, 6, 4, 5], [2, 3, 4, 1, 6, 5],

 [2, 3, 4, 5, 6, 1], [2, 3, 4, 6, 1, 5], [2, 3, 4, 6, 5, 1], [2, 3, 5, 4, 6, 1],

 [2, 3, 5, 6, 4, 1], [2, 3, 6, 1, 4, 5], [2, 3, 6, 4, 1, 5], [2, 3, 6, 4, 5, 1],

 [2, 3, 6, 5, 4, 1], [2, 4, 1, 3, 6, 5], [2, 4, 1, 5, 6, 3], [2, 4, 1, 6, 3, 5],

 [2, 4, 1, 6, 5, 3], [2, 4, 3, 1, 6, 5], [2, 4, 3, 5, 6, 1], [2, 4, 3, 6, 1, 5],

 [2, 4, 3, 6, 5, 1], [2, 4, 5, 1, 6, 3], [2, 4, 5, 3, 6, 1], [2, 4, 5, 6, 1, 3],

 [2, 4, 5, 6, 3, 1], [2, 4, 6, 1, 3, 5], [2, 4, 6, 1, 5, 3], [2, 4, 6, 3, 1, 5],

 [2, 4, 6, 3, 5, 1], [2, 4, 6, 5, 1, 3], [2, 4, 6, 5, 3, 1], [2, 5, 1, 4, 6, 3],

 [2, 5, 1, 6, 4, 3], [2, 5, 3, 4, 6, 1], [2, 5, 3, 6, 4, 1], [2, 5, 4, 1, 6, 3],

 [2, 5, 4, 3, 6, 1], [2, 5, 4, 6, 1, 3], [2, 5, 4, 6, 3, 1], [2, 5, 6, 1, 4, 3],

 [2, 5, 6, 3, 4, 1], [2, 5, 6, 4, 1, 3], [2, 5, 6, 4, 3, 1], [2, 6, 1, 3, 4, 5],

 [2, 6, 1, 4, 3, 5], [2, 6, 1, 4, 5, 3], [2, 6, 1, 5, 4, 3], [2, 6, 3, 1, 4, 5],

 [2, 6, 3, 4, 1, 5], [2, 6, 3, 4, 5, 1], [2, 6, 3, 5, 4, 1], [2, 6, 4, 1, 3, 5],

 [2, 6, 4, 1, 5, 3], [2, 6, 4, 3, 1, 5], [2, 6, 4, 3, 5, 1], [2, 6, 4, 5, 1, 3],

 [2, 6, 4, 5, 3, 1], [2, 6, 5, 1, 4, 3], [2, 6, 5, 3, 4, 1], [2, 6, 5, 4, 1, 3],

 [2, 6, 5, 4, 3, 1], [3, 1, 2, 4, 6, 5], [3, 1, 2, 6, 4, 5], [3, 1, 4, 2, 6, 5],

 [3, 1, 4, 6, 2, 5], [3, 1, 6, 2, 4, 5], [3, 1, 6, 4, 2, 5], [3, 2, 1, 4, 6, 5],

 [3, 2, 1, 6, 4, 5], [3, 2, 4, 1, 6, 5], [3, 2, 4, 5, 6, 1], [3, 2, 4, 6, 1, 5],

 [3, 2, 4, 6, 5, 1], [3, 2, 5, 4, 6, 1], [3, 2, 5, 6, 4, 1], [3, 2, 6, 1, 4, 5],

 [3, 2, 6, 4, 1, 5], [3, 2, 6, 4, 5, 1], [3, 2, 6, 5, 4, 1], [3, 4, 1, 2, 6, 5],

 [3, 4, 1, 6, 2, 5], [3, 4, 2, 1, 6, 5], [3, 4, 2, 5, 6, 1], [3, 4, 2, 6, 1, 5],

 [3, 4, 2, 6, 5, 1], [3, 4, 5, 2, 6, 1], [3, 4, 5, 6, 2, 1], [3, 4, 6, 1, 2, 5],

 [3, 4, 6, 2, 1, 5], [3, 4, 6, 2, 5, 1], [3, 4, 6, 5, 2, 1], [3, 5, 2, 4, 6, 1],

 [3, 5, 2, 6, 4, 1], [3, 5, 4, 2, 6, 1], [3, 5, 4, 6, 2, 1], [3, 5, 6, 2, 4, 1],

 [3, 5, 6, 4, 2, 1], [3, 6, 1, 2, 4, 5], [3, 6, 1, 4, 2, 5], [3, 6, 2, 1, 4, 5],

 [3, 6, 2, 4, 1, 5], [3, 6, 2, 4, 5, 1], [3, 6, 2, 5, 4, 1], [3, 6, 4, 1, 2, 5],

 [3, 6, 4, 2, 1, 5], [3, 6, 4, 2, 5, 1], [3, 6, 4, 5, 2, 1], [3, 6, 5, 2, 4, 1],

 [3, 6, 5, 4, 2, 1], [4, 1, 2, 3, 6, 5], [4, 1, 2, 5, 6, 3], [4, 1, 2, 6, 3, 5],

 [4, 1, 2, 6, 5, 3], [4, 1, 3, 2, 6, 5], [4, 1, 3, 6, 2, 5], [4, 1, 5, 2, 6, 3],

 [4, 1, 5, 6, 2, 3], [4, 1, 6, 2, 3, 5], [4, 1, 6, 2, 5, 3], [4, 1, 6, 3, 2, 5],

 [4, 1, 6, 5, 2, 3], [4, 2, 1, 3, 6, 5], [4, 2, 1, 5, 6, 3], [4, 2, 1, 6, 3, 5],

 [4, 2, 1, 6, 5, 3], [4, 2, 3, 1, 6, 5], [4, 2, 3, 5, 6, 1], [4, 2, 3, 6, 1, 5],

 [4, 2, 3, 6, 5, 1], [4, 2, 5, 1, 6, 3], [4, 2, 5, 3, 6, 1], [4, 2, 5, 6, 1, 3],

 [4, 2, 5, 6, 3, 1], [4, 2, 6, 1, 3, 5], [4, 2, 6, 1, 5, 3], [4, 2, 6, 3, 1, 5],

 [4, 2, 6, 3, 5, 1], [4, 2, 6, 5, 1, 3], [4, 2, 6, 5, 3, 1], [4, 3, 1, 2, 6, 5],

 [4, 3, 1, 6, 2, 5], [4, 3, 2, 1, 6, 5], [4, 3, 2, 5, 6, 1], [4, 3, 2, 6, 1, 5],

 [4, 3, 2, 6, 5, 1], [4, 3, 5, 2, 6, 1], [4, 3, 5, 6, 2, 1], [4, 3, 6, 1, 2, 5],

 [4, 3, 6, 2, 1, 5], [4, 3, 6, 2, 5, 1], [4, 3, 6, 5, 2, 1], [4, 5, 1, 2, 6, 3],

 [4, 5, 1, 6, 2, 3], [4, 5, 2, 1, 6, 3], [4, 5, 2, 3, 6, 1], [4, 5, 2, 6, 1, 3],

 [4, 5, 2, 6, 3, 1], [4, 5, 3, 2, 6, 1], [4, 5, 3, 6, 2, 1], [4, 5, 6, 1, 2, 3],

 [4, 5, 6, 2, 1, 3], [4, 5, 6, 2, 3, 1], [4, 5, 6, 3, 2, 1], [4, 6, 1, 2, 3, 5],

 [4, 6, 1, 2, 5, 3], [4, 6, 1, 3, 2, 5], [4, 6, 1, 5, 2, 3], [4, 6, 2, 1, 3, 5],

 [4, 6, 2, 1, 5, 3], [4, 6, 2, 3, 1, 5], [4, 6, 2, 3, 5, 1], [4, 6, 2, 5, 1, 3],

 [4, 6, 2, 5, 3, 1], [4, 6, 3, 1, 2, 5], [4, 6, 3, 2, 1, 5], [4, 6, 3, 2, 5, 1],

 [4, 6, 3, 5, 2, 1], [4, 6, 5, 1, 2, 3], [4, 6, 5, 2, 1, 3], [4, 6, 5, 2, 3, 1],

 [4, 6, 5, 3, 2, 1], [5, 1, 2, 4, 6, 3], [5, 1, 2, 6, 4, 3], [5, 1, 4, 2, 6, 3],

 [5, 1, 4, 6, 2, 3], [5, 1, 6, 2, 4, 3], [5, 1, 6, 4, 2, 3], [5, 2, 1, 4, 6, 3],

 [5, 2, 1, 6, 4, 3], [5, 2, 3, 4, 6, 1], [5, 2, 3, 6, 4, 1], [5, 2, 4, 1, 6, 3],

 [5, 2, 4, 3, 6, 1], [5, 2, 4, 6, 1, 3], [5, 2, 4, 6, 3, 1], [5, 2, 6, 1, 4, 3],

 [5, 2, 6, 3, 4, 1], [5, 2, 6, 4, 1, 3], [5, 2, 6, 4, 3, 1], [5, 3, 2, 4, 6, 1],

 [5, 3, 2, 6, 4, 1], [5, 3, 4, 2, 6, 1], [5, 3, 4, 6, 2, 1], [5, 3, 6, 2, 4, 1],

 [5, 3, 6, 4, 2, 1], [5, 4, 1, 2, 6, 3], [5, 4, 1, 6, 2, 3], [5, 4, 2, 1, 6, 3],

 [5, 4, 2, 3, 6, 1], [5, 4, 2, 6, 1, 3], [5, 4, 2, 6, 3, 1], [5, 4, 3, 2, 6, 1],

 [5, 4, 3, 6, 2, 1], [5, 4, 6, 1, 2, 3], [5, 4, 6, 2, 1, 3], [5, 4, 6, 2, 3, 1],

 [5, 4, 6, 3, 2, 1], [5, 6, 1, 2, 4, 3], [5, 6, 1, 4, 2, 3], [5, 6, 2, 1, 4, 3],

 [5, 6, 2, 3, 4, 1], [5, 6, 2, 4, 1, 3], [5, 6, 2, 4, 3, 1], [5, 6, 3, 2, 4, 1],

 [5, 6, 3, 4, 2, 1], [5, 6, 4, 1, 2, 3], [5, 6, 4, 2, 1, 3], [5, 6, 4, 2, 3, 1],

 [5, 6, 4, 3, 2, 1], [6, 1, 2, 3, 4, 5], [6, 1, 2, 4, 3, 5], [6, 1, 2, 4, 5, 3],

 [6, 1, 2, 5, 4, 3], [6, 1, 3, 2, 4, 5], [6, 1, 3, 4, 2, 5], [6, 1, 4, 2, 3, 5],

 [6, 1, 4, 2, 5, 3], [6, 1, 4, 3, 2, 5], [6, 1, 4, 5, 2, 3], [6, 1, 5, 2, 4, 3],

 [6, 1, 5, 4, 2, 3], [6, 2, 1, 3, 4, 5], [6, 2, 1, 4, 3, 5], [6, 2, 1, 4, 5, 3],

 [6, 2, 1, 5, 4, 3], [6, 2, 3, 1, 4, 5], [6, 2, 3, 4, 1, 5], [6, 2, 3, 4, 5, 1],

 [6, 2, 3, 5, 4, 1], [6, 2, 4, 1, 3, 5], [6, 2, 4, 1, 5, 3], [6, 2, 4, 3, 1, 5],

 [6, 2, 4, 3, 5, 1], [6, 2, 4, 5, 1, 3], [6, 2, 4, 5, 3, 1], [6, 2, 5, 1, 4, 3],

 [6, 2, 5, 3, 4, 1], [6, 2, 5, 4, 1, 3], [6, 2, 5, 4, 3, 1], [6, 3, 1, 2, 4, 5],

 [6, 3, 1, 4, 2, 5], [6, 3, 2, 1, 4, 5], [6, 3, 2, 4, 1, 5], [6, 3, 2, 4, 5, 1],

 [6, 3, 2, 5, 4, 1], [6, 3, 4, 1, 2, 5], [6, 3, 4, 2, 1, 5], [6, 3, 4, 2, 5, 1],

 [6, 3, 4, 5, 2, 1], [6, 3, 5, 2, 4, 1], [6, 3, 5, 4, 2, 1], [6, 4, 1, 2, 3, 5],

 [6, 4, 1, 2, 5, 3], [6, 4, 1, 3, 2, 5], [6, 4, 1, 5, 2, 3], [6, 4, 2, 1, 3, 5],

 [6, 4, 2, 1, 5, 3], [6, 4, 2, 3, 1, 5], [6, 4, 2, 3, 5, 1], [6, 4, 2, 5, 1, 3],

 [6, 4, 2, 5, 3, 1], [6, 4, 3, 1, 2, 5], [6, 4, 3, 2, 1, 5], [6, 4, 3, 2, 5, 1],

 [6, 4, 3, 5, 2, 1], [6, 4, 5, 1, 2, 3], [6, 4, 5, 2, 1, 3], [6, 4, 5, 2, 3, 1],

 [6, 4, 5, 3, 2, 1], [6, 5, 1, 2, 4, 3], [6, 5, 1, 4, 2, 3], [6, 5, 2, 1, 4, 3],

 [6, 5, 2, 3, 4, 1], [6, 5, 2, 4, 1, 3], [6, 5, 2, 4, 3, 1], [6, 5, 3, 2, 4, 1],

 [6, 5, 3, 4, 2, 1], [6, 5, 4, 1, 2, 3], [6, 5, 4, 2, 1, 3], [6, 5, 4, 2, 3, 1],

 [6, 5, 4, 3, 2, 1] ]

d) dio

len(list(filter(lambda x: x[-2]==2 and x[-1]==1,permS)))

24

list(filter(lambda x: x[-2]==2 and x[-1]==1,permS))

[ [3, 4, 5, 6, 2, 1], [3, 4, 6, 5, 2, 1], [3, 5, 4, 6, 2, 1],

 [3, 5, 6, 4, 2, 1], [3, 6, 4, 5, 2, 1], [3, 6, 5, 4, 2, 1], [4, 3, 5, 6, 2, 1],

 [4, 3, 6, 5, 2, 1], [4, 5, 3, 6, 2, 1], [4, 5, 6, 3, 2, 1], [4, 6, 3, 5, 2, 1],

 [4, 6, 5, 3, 2, 1], [5, 3, 4, 6, 2, 1], [5, 3, 6, 4, 2, 1], [5, 4, 3, 6, 2, 1],

 [5, 4, 6, 3, 2, 1], [5, 6, 3, 4, 2, 1], [5, 6, 4, 3, 2, 1], [6, 3, 4, 5, 2, 1],

 [6, 3, 5, 4, 2, 1], [6, 4, 3, 5, 2, 1], [6, 4, 5, 3, 2, 1], [6, 5, 3, 4, 2, 1],

 [6, 5, 4, 3, 2, 1] ]

e) dio

len(list(filter(lambda x: not(x[-3]==2 and x[-2]==3 and x[-1]==4),permS)))

714

list(filter(lambda x: not(x[-3]==2 and x[-2]==3 and x[-1]==4),permS))

[ [1, 2, 3, 4, 5, 6], [1, 2, 3, 4, 6, 5], [1, 2, 3, 5, 4, 6],

 [1, 2, 3, 5, 6, 4], [1, 2, 3, 6, 4, 5], [1, 2, 3, 6, 5, 4], [1, 2, 4, 3, 5, 6],

 [1, 2, 4, 3, 6, 5], [1, 2, 4, 5, 3, 6], [1, 2, 4, 5, 6, 3], [1, 2, 4, 6, 3, 5],

 [1, 2, 4, 6, 5, 3], [1, 2, 5, 3, 4, 6], [1, 2, 5, 3, 6, 4], [1, 2, 5, 4, 3, 6],

 [1, 2, 5, 4, 6, 3], [1, 2, 5, 6, 3, 4], [1, 2, 5, 6, 4, 3], [1, 2, 6, 3, 4, 5],

 [1, 2, 6, 3, 5, 4], [1, 2, 6, 4, 3, 5], [1, 2, 6, 4, 5, 3], [1, 2, 6, 5, 3, 4],

 [1, 2, 6, 5, 4, 3], [1, 3, 2, 4, 5, 6], [1, 3, 2, 4, 6, 5], [1, 3, 2, 5, 4, 6],

 [1, 3, 2, 5, 6, 4], [1, 3, 2, 6, 4, 5], [1, 3, 2, 6, 5, 4], [1, 3, 4, 2, 5, 6],

 [1, 3, 4, 2, 6, 5], [1, 3, 4, 5, 2, 6], [1, 3, 4, 5, 6, 2], [1, 3, 4, 6, 2, 5],

 [1, 3, 4, 6, 5, 2], [1, 3, 5, 2, 4, 6], [1, 3, 5, 2, 6, 4], [1, 3, 5, 4, 2, 6],

 [1, 3, 5, 4, 6, 2], [1, 3, 5, 6, 2, 4], [1, 3, 5, 6, 4, 2], [1, 3, 6, 2, 4, 5],

 [1, 3, 6, 2, 5, 4], [1, 3, 6, 4, 2, 5], [1, 3, 6, 4, 5, 2], [1, 3, 6, 5, 2, 4],

 [1, 3, 6, 5, 4, 2], [1, 4, 2, 3, 5, 6], [1, 4, 2, 3, 6, 5], [1, 4, 2, 5, 3, 6],

 [1, 4, 2, 5, 6, 3], [1, 4, 2, 6, 3, 5], [1, 4, 2, 6, 5, 3], [1, 4, 3, 2, 5, 6],

 [1, 4, 3, 2, 6, 5], [1, 4, 3, 5, 2, 6], [1, 4, 3, 5, 6, 2], [1, 4, 3, 6, 2, 5],

 [1, 4, 3, 6, 5, 2], [1, 4, 5, 2, 3, 6], [1, 4, 5, 2, 6, 3], [1, 4, 5, 3, 2, 6],

 [1, 4, 5, 3, 6, 2], [1, 4, 5, 6, 2, 3], [1, 4, 5, 6, 3, 2], [1, 4, 6, 2, 3, 5],

 [1, 4, 6, 2, 5, 3], [1, 4, 6, 3, 2, 5], [1, 4, 6, 3, 5, 2], [1, 4, 6, 5, 2, 3],

 [1, 4, 6, 5, 3, 2], [1, 5, 2, 3, 4, 6], [1, 5, 2, 3, 6, 4], [1, 5, 2, 4, 3, 6],

 [1, 5, 2, 4, 6, 3], [1, 5, 2, 6, 3, 4], [1, 5, 2, 6, 4, 3], [1, 5, 3, 2, 4, 6],

 [1, 5, 3, 2, 6, 4], [1, 5, 3, 4, 2, 6], [1, 5, 3, 4, 6, 2], [1, 5, 3, 6, 2, 4],

 [1, 5, 3, 6, 4, 2], [1, 5, 4, 2, 3, 6], [1, 5, 4, 2, 6, 3], [1, 5, 4, 3, 2, 6],

 [1, 5, 4, 3, 6, 2], [1, 5, 4, 6, 2, 3], [1, 5, 4, 6, 3, 2], [1, 5, 6, 2, 4, 3],

 [1, 5, 6, 3, 2, 4], [1, 5, 6, 3, 4, 2], [1, 5, 6, 4, 2, 3], [1, 5, 6, 4, 3, 2],

 [1, 6, 2, 3, 4, 5], [1, 6, 2, 3, 5, 4], [1, 6, 2, 4, 3, 5], [1, 6, 2, 4, 5, 3],

 [1, 6, 2, 5, 3, 4], [1, 6, 2, 5, 4, 3], [1, 6, 3, 2, 4, 5], [1, 6, 3, 2, 5, 4],

 [1, 6, 3, 4, 2, 5], [1, 6, 3, 4, 5, 2], [1, 6, 3, 5, 2, 4], [1, 6, 3, 5, 4, 2],

 [1, 6, 4, 2, 3, 5], [1, 6, 4, 2, 5, 3], [1, 6, 4, 3, 2, 5], [1, 6, 4, 3, 5, 2],

 [1, 6, 4, 5, 2, 3], [1, 6, 4, 5, 3, 2], [1, 6, 5, 2, 4, 3], [1, 6, 5, 3, 2, 4],

 [1, 6, 5, 3, 4, 2], [1, 6, 5, 4, 2, 3], [1, 6, 5, 4, 3, 2], [2, 1, 3, 4, 5, 6],

 [2, 1, 3, 4, 6, 5], [2, 1, 3, 5, 4, 6], [2, 1, 3, 5, 6, 4], [2, 1, 3, 6, 4, 5],

 [2, 1, 3, 6, 5, 4], [2, 1, 4, 3, 5, 6], [2, 1, 4, 3, 6, 5], [2, 1, 4, 5, 3, 6],

 [2, 1, 4, 5, 6, 3], [2, 1, 4, 6, 3, 5], [2, 1, 4, 6, 5, 3], [2, 1, 5, 3, 4, 6],

 [2, 1, 5, 3, 6, 4], [2, 1, 5, 4, 3, 6], [2, 1, 5, 4, 6, 3], [2, 1, 5, 6, 3, 4],

 [2, 1, 5, 6, 4, 3], [2, 1, 6, 3, 4, 5], [2, 1, 6, 3, 5, 4], [2, 1, 6, 4, 3, 5],

 [2, 1, 6, 4, 5, 3], [2, 1, 6, 5, 3, 4], [2, 1, 6, 5, 4, 3], [2, 3, 1, 4, 5, 6],

 [2, 3, 1, 4, 6, 5], [2, 3, 1, 5, 4, 6], [2, 3, 1, 5, 6, 4], [2, 3, 1, 6, 4, 5],

 [2, 3, 1, 6, 5, 4], [2, 3, 4, 1, 5, 6], [2, 3, 4, 1, 6, 5], [2, 3, 4, 5, 1, 6],

 [2, 3, 4, 5, 6, 1], [2, 3, 4, 6, 1, 5], [2, 3, 4, 6, 5, 1], [2, 3, 5, 1, 4, 6],

 [2, 3, 5, 1, 6, 4], [2, 3, 5, 4, 1, 6], [2, 3, 5, 4, 6, 1], [2, 3, 5, 6, 1, 4],

 [2, 3, 5, 6, 4, 1], [2, 3, 6, 1, 4, 5], [2, 3, 6, 1, 5, 4], [2, 3, 6, 4, 1, 5],

 [2, 3, 6, 4, 5, 1], [2, 3, 6, 5, 1, 4], [2, 3, 6, 5, 4, 1], [2, 4, 1, 3, 5, 6],

 [2, 4, 1, 3, 6, 5], [2, 4, 1, 5, 3, 6], [2, 4, 1, 5, 6, 3], [2, 4, 1, 6, 3, 5],

 [2, 4, 1, 6, 5, 3], [2, 4, 3, 1, 5, 6], [2, 4, 3, 1, 6, 5], [2, 4, 3, 5, 1, 6],

 [2, 4, 3, 5, 6, 1], [2, 4, 3, 6, 1, 5], [2, 4, 3, 6, 5, 1], [2, 4, 5, 1, 3, 6],

 [2, 4, 5, 1, 6, 3], [2, 4, 5, 3, 1, 6], [2, 4, 5, 3, 6, 1], [2, 4, 5, 6, 1, 3],

 [2, 4, 5, 6, 3, 1], [2, 4, 6, 1, 3, 5], [2, 4, 6, 1, 5, 3], [2, 4, 6, 3, 1, 5],

 [2, 4, 6, 3, 5, 1], [2, 4, 6, 5, 1, 3], [2, 4, 6, 5, 3, 1], [2, 5, 1, 3, 4, 6],

 [2, 5, 1, 3, 6, 4], [2, 5, 1, 4, 3, 6], [2, 5, 1, 4, 6, 3], [2, 5, 1, 6, 3, 4],

 [2, 5, 1, 6, 4, 3], [2, 5, 3, 1, 4, 6], [2, 5, 3, 1, 6, 4], [2, 5, 3, 4, 1, 6],

 [2, 5, 3, 4, 6, 1], [2, 5, 3, 6, 1, 4], [2, 5, 3, 6, 4, 1], [2, 5, 4, 1, 3, 6],

 [2, 5, 4, 1, 6, 3], [2, 5, 4, 3, 1, 6], [2, 5, 4, 3, 6, 1], [2, 5, 4, 6, 1, 3],

 [2, 5, 4, 6, 3, 1], [2, 5, 6, 1, 3, 4], [2, 5, 6, 1, 4, 3], [2, 5, 6, 3, 1, 4],

 [2, 5, 6, 3, 4, 1], [2, 5, 6, 4, 1, 3], [2, 5, 6, 4, 3, 1], [2, 6, 1, 3, 4, 5],

 [2, 6, 1, 3, 5, 4], [2, 6, 1, 4, 3, 5], [2, 6, 1, 4, 5, 3], [2, 6, 1, 5, 3, 4],

 [2, 6, 1, 5, 4, 3], [2, 6, 3, 1, 4, 5], [2, 6, 3, 1, 5, 4], [2, 6, 3, 4, 1, 5],

 [2, 6, 3, 4, 5, 1], [2, 6, 3, 5, 1, 4], [2, 6, 3, 5, 4, 1], [2, 6, 4, 1, 3, 5],

 [2, 6, 4, 1, 5, 3], [2, 6, 4, 3, 1, 5], [2, 6, 4, 3, 5, 1], [2, 6, 4, 5, 1, 3],

 [2, 6, 4, 5, 3, 1], [2, 6, 5, 1, 3, 4], [2, 6, 5, 1, 4, 3], [2, 6, 5, 3, 1, 4],

 [2, 6, 5, 3, 4, 1], [2, 6, 5, 4, 1, 3], [2, 6, 5, 4, 3, 1], [3, 1, 2, 4, 5, 6],

 [3, 1, 2, 4, 6, 5], [3, 1, 2, 5, 4, 6], [3, 1, 2, 5, 6, 4], [3, 1, 2, 6, 4, 5],

 [3, 1, 2, 6, 5, 4], [3, 1, 4, 2, 5, 6], [3, 1, 4, 2, 6, 5], [3, 1, 4, 5, 2, 6],

 [3, 1, 4, 5, 6, 2], [3, 1, 4, 6, 2, 5], [3, 1, 4, 6, 5, 2], [3, 1, 5, 2, 4, 6],

 [3, 1, 5, 2, 6, 4], [3, 1, 5, 4, 2, 6], [3, 1, 5, 4, 6, 2], [3, 1, 5, 6, 2, 4],

 [3, 1, 5, 6, 4, 2], [3, 1, 6, 2, 4, 5], [3, 1, 6, 2, 5, 4], [3, 1, 6, 4, 2, 5],

 [3, 1, 6, 4, 5, 2], [3, 1, 6, 5, 2, 4], [3, 1, 6, 5, 4, 2], [3, 2, 1, 4, 5, 6],

 [3, 2, 1, 4, 6, 5], [3, 2, 1, 5, 4, 6], [3, 2, 1, 5, 6, 4], [3, 2, 1, 6, 4, 5],

 [3, 2, 1, 6, 5, 4], [3, 2, 4, 1, 5, 6], [3, 2, 4, 1, 6, 5], [3, 2, 4, 5, 1, 6],

 [3, 2, 4, 5, 6, 1], [3, 2, 4, 6, 1, 5], [3, 2, 4, 6, 5, 1], [3, 2, 5, 1, 4, 6],

 [3, 2, 5, 1, 6, 4], [3, 2, 5, 4, 1, 6], [3, 2, 5, 4, 6, 1], [3, 2, 5, 6, 1, 4],

 [3, 2, 5, 6, 4, 1], [3, 2, 6, 1, 4, 5], [3, 2, 6, 1, 5, 4], [3, 2, 6, 4, 1, 5],

 [3, 2, 6, 4, 5, 1], [3, 2, 6, 5, 1, 4], [3, 2, 6, 5, 4, 1], [3, 4, 1, 2, 5, 6],

 [3, 4, 1, 2, 6, 5], [3, 4, 1, 5, 2, 6], [3, 4, 1, 5, 6, 2], [3, 4, 1, 6, 2, 5],

 [3, 4, 1, 6, 5, 2], [3, 4, 2, 1, 5, 6], [3, 4, 2, 1, 6, 5], [3, 4, 2, 5, 1, 6],

 [3, 4, 2, 5, 6, 1], [3, 4, 2, 6, 1, 5], [3, 4, 2, 6, 5, 1], [3, 4, 5, 1, 2, 6],

 [3, 4, 5, 1, 6, 2], [3, 4, 5, 2, 1, 6], [3, 4, 5, 2, 6, 1], [3, 4, 5, 6, 1, 2],

 [3, 4, 5, 6, 2, 1], [3, 4, 6, 1, 2, 5], [3, 4, 6, 1, 5, 2], [3, 4, 6, 2, 1, 5],

 [3, 4, 6, 2, 5, 1], [3, 4, 6, 5, 1, 2], [3, 4, 6, 5, 2, 1], [3, 5, 1, 2, 4, 6],

 [3, 5, 1, 2, 6, 4], [3, 5, 1, 4, 2, 6], [3, 5, 1, 4, 6, 2], [3, 5, 1, 6, 2, 4],

 [3, 5, 1, 6, 4, 2], [3, 5, 2, 1, 4, 6], [3, 5, 2, 1, 6, 4], [3, 5, 2, 4, 1, 6],

 [3, 5, 2, 4, 6, 1], [3, 5, 2, 6, 1, 4], [3, 5, 2, 6, 4, 1], [3, 5, 4, 1, 2, 6],

 [3, 5, 4, 1, 6, 2], [3, 5, 4, 2, 1, 6], [3, 5, 4, 2, 6, 1], [3, 5, 4, 6, 1, 2],

 [3, 5, 4, 6, 2, 1], [3, 5, 6, 1, 2, 4], [3, 5, 6, 1, 4, 2], [3, 5, 6, 2, 1, 4],

 [3, 5, 6, 2, 4, 1], [3, 5, 6, 4, 1, 2], [3, 5, 6, 4, 2, 1], [3, 6, 1, 2, 4, 5],

 [3, 6, 1, 2, 5, 4], [3, 6, 1, 4, 2, 5], [3, 6, 1, 4, 5, 2], [3, 6, 1, 5, 2, 4],

 [3, 6, 1, 5, 4, 2], [3, 6, 2, 1, 4, 5], [3, 6, 2, 1, 5, 4], [3, 6, 2, 4, 1, 5],

 [3, 6, 2, 4, 5, 1], [3, 6, 2, 5, 1, 4], [3, 6, 2, 5, 4, 1], [3, 6, 4, 1, 2, 5],

 [3, 6, 4, 1, 5, 2], [3, 6, 4, 2, 1, 5], [3, 6, 4, 2, 5, 1], [3, 6, 4, 5, 1, 2],

 [3, 6, 4, 5, 2, 1], [3, 6, 5, 1, 2, 4], [3, 6, 5, 1, 4, 2], [3, 6, 5, 2, 1, 4],

 [3, 6, 5, 2, 4, 1], [3, 6, 5, 4, 1, 2], [3, 6, 5, 4, 2, 1], [4, 1, 2, 3, 5, 6],

 [4, 1, 2, 3, 6, 5], [4, 1, 2, 5, 3, 6], [4, 1, 2, 5, 6, 3], [4, 1, 2, 6, 3, 5],

 [4, 1, 2, 6, 5, 3], [4, 1, 3, 2, 5, 6], [4, 1, 3, 2, 6, 5], [4, 1, 3, 5, 2, 6],

 [4, 1, 3, 5, 6, 2], [4, 1, 3, 6, 2, 5], [4, 1, 3, 6, 5, 2], [4, 1, 5, 2, 3, 6],

 [4, 1, 5, 2, 6, 3], [4, 1, 5, 3, 2, 6], [4, 1, 5, 3, 6, 2], [4, 1, 5, 6, 2, 3],

 [4, 1, 5, 6, 3, 2], [4, 1, 6, 2, 3, 5], [4, 1, 6, 2, 5, 3], [4, 1, 6, 3, 2, 5],

 [4, 1, 6, 3, 5, 2], [4, 1, 6, 5, 2, 3], [4, 1, 6, 5, 3, 2], [4, 2, 1, 3, 5, 6],

 [4, 2, 1, 3, 6, 5], [4, 2, 1, 5, 3, 6], [4, 2, 1, 5, 6, 3], [4, 2, 1, 6, 3, 5],

 [4, 2, 1, 6, 5, 3], [4, 2, 3, 1, 5, 6], [4, 2, 3, 1, 6, 5], [4, 2, 3, 5, 1, 6],

 [4, 2, 3, 5, 6, 1], [4, 2, 3, 6, 1, 5], [4, 2, 3, 6, 5, 1], [4, 2, 5, 1, 3, 6],

 [4, 2, 5, 1, 6, 3], [4, 2, 5, 3, 1, 6], [4, 2, 5, 3, 6, 1], [4, 2, 5, 6, 1, 3],

 [4, 2, 5, 6, 3, 1], [4, 2, 6, 1, 3, 5], [4, 2, 6, 1, 5, 3], [4, 2, 6, 3, 1, 5],

 [4, 2, 6, 3, 5, 1], [4, 2, 6, 5, 1, 3], [4, 2, 6, 5, 3, 1], [4, 3, 1, 2, 5, 6],

 [4, 3, 1, 2, 6, 5], [4, 3, 1, 5, 2, 6], [4, 3, 1, 5, 6, 2], [4, 3, 1, 6, 2, 5],

 [4, 3, 1, 6, 5, 2], [4, 3, 2, 1, 5, 6], [4, 3, 2, 1, 6, 5], [4, 3, 2, 5, 1, 6],

 [4, 3, 2, 5, 6, 1], [4, 3, 2, 6, 1, 5], [4, 3, 2, 6, 5, 1], [4, 3, 5, 1, 2, 6],

 [4, 3, 5, 1, 6, 2], [4, 3, 5, 2, 1, 6], [4, 3, 5, 2, 6, 1], [4, 3, 5, 6, 1, 2],

 [4, 3, 5, 6, 2, 1], [4, 3, 6, 1, 2, 5], [4, 3, 6, 1, 5, 2], [4, 3, 6, 2, 1, 5],

 [4, 3, 6, 2, 5, 1], [4, 3, 6, 5, 1, 2], [4, 3, 6, 5, 2, 1], [4, 5, 1, 2, 3, 6],

 [4, 5, 1, 2, 6, 3], [4, 5, 1, 3, 2, 6], [4, 5, 1, 3, 6, 2], [4, 5, 1, 6, 2, 3],

 [4, 5, 1, 6, 3, 2], [4, 5, 2, 1, 3, 6], [4, 5, 2, 1, 6, 3], [4, 5, 2, 3, 1, 6],

 [4, 5, 2, 3, 6, 1], [4, 5, 2, 6, 1, 3], [4, 5, 2, 6, 3, 1], [4, 5, 3, 1, 2, 6],

 [4, 5, 3, 1, 6, 2], [4, 5, 3, 2, 1, 6], [4, 5, 3, 2, 6, 1], [4, 5, 3, 6, 1, 2],

 [4, 5, 3, 6, 2, 1], [4, 5, 6, 1, 2, 3], [4, 5, 6, 1, 3, 2], [4, 5, 6, 2, 1, 3],

 [4, 5, 6, 2, 3, 1], [4, 5, 6, 3, 1, 2], [4, 5, 6, 3, 2, 1], [4, 6, 1, 2, 3, 5],

 [4, 6, 1, 2, 5, 3], [4, 6, 1, 3, 2, 5], [4, 6, 1, 3, 5, 2], [4, 6, 1, 5, 2, 3],

 [4, 6, 1, 5, 3, 2], [4, 6, 2, 1, 3, 5], [4, 6, 2, 1, 5, 3], [4, 6, 2, 3, 1, 5],

 [4, 6, 2, 3, 5, 1], [4, 6, 2, 5, 1, 3], [4, 6, 2, 5, 3, 1], [4, 6, 3, 1, 2, 5],

 [4, 6, 3, 1, 5, 2], [4, 6, 3, 2, 1, 5], [4, 6, 3, 2, 5, 1], [4, 6, 3, 5, 1, 2],

 [4, 6, 3, 5, 2, 1], [4, 6, 5, 1, 2, 3], [4, 6, 5, 1, 3, 2], [4, 6, 5, 2, 1, 3],

 [4, 6, 5, 2, 3, 1], [4, 6, 5, 3, 1, 2], [4, 6, 5, 3, 2, 1], [5, 1, 2, 3, 4, 6],

 [5, 1, 2, 3, 6, 4], [5, 1, 2, 4, 3, 6], [5, 1, 2, 4, 6, 3], [5, 1, 2, 6, 3, 4],

 [5, 1, 2, 6, 4, 3], [5, 1, 3, 2, 4, 6], [5, 1, 3, 2, 6, 4], [5, 1, 3, 4, 2, 6],

 [5, 1, 3, 4, 6, 2], [5, 1, 3, 6, 2, 4], [5, 1, 3, 6, 4, 2], [5, 1, 4, 2, 3, 6],

 [5, 1, 4, 2, 6, 3], [5, 1, 4, 3, 2, 6], [5, 1, 4, 3, 6, 2], [5, 1, 4, 6, 2, 3],

 [5, 1, 4, 6, 3, 2], [5, 1, 6, 2, 4, 3], [5, 1, 6, 3, 2, 4], [5, 1, 6, 3, 4, 2],

 [5, 1, 6, 4, 2, 3], [5, 1, 6, 4, 3, 2], [5, 2, 1, 3, 4, 6], [5, 2, 1, 3, 6, 4],

 [5, 2, 1, 4, 3, 6], [5, 2, 1, 4, 6, 3], [5, 2, 1, 6, 3, 4], [5, 2, 1, 6, 4, 3],

 [5, 2, 3, 1, 4, 6], [5, 2, 3, 1, 6, 4], [5, 2, 3, 4, 1, 6], [5, 2, 3, 4, 6, 1],

 [5, 2, 3, 6, 1, 4], [5, 2, 3, 6, 4, 1], [5, 2, 4, 1, 3, 6], [5, 2, 4, 1, 6, 3],

 [5, 2, 4, 3, 1, 6], [5, 2, 4, 3, 6, 1], [5, 2, 4, 6, 1, 3], [5, 2, 4, 6, 3, 1],

 [5, 2, 6, 1, 3, 4], [5, 2, 6, 1, 4, 3], [5, 2, 6, 3, 1, 4], [5, 2, 6, 3, 4, 1],

 [5, 2, 6, 4, 1, 3], [5, 2, 6, 4, 3, 1], [5, 3, 1, 2, 4, 6], [5, 3, 1, 2, 6, 4],

 [5, 3, 1, 4, 2, 6], [5, 3, 1, 4, 6, 2], [5, 3, 1, 6, 2, 4], [5, 3, 1, 6, 4, 2],

 [5, 3, 2, 1, 4, 6], [5, 3, 2, 1, 6, 4], [5, 3, 2, 4, 1, 6], [5, 3, 2, 4, 6, 1],

 [5, 3, 2, 6, 1, 4], [5, 3, 2, 6, 4, 1], [5, 3, 4, 1, 2, 6], [5, 3, 4, 1, 6, 2],

 [5, 3, 4, 2, 1, 6], [5, 3, 4, 2, 6, 1], [5, 3, 4, 6, 1, 2], [5, 3, 4, 6, 2, 1],

 [5, 3, 6, 1, 2, 4], [5, 3, 6, 1, 4, 2], [5, 3, 6, 2, 1, 4], [5, 3, 6, 2, 4, 1],

 [5, 3, 6, 4, 1, 2], [5, 3, 6, 4, 2, 1], [5, 4, 1, 2, 3, 6], [5, 4, 1, 2, 6, 3],

 [5, 4, 1, 3, 2, 6], [5, 4, 1, 3, 6, 2], [5, 4, 1, 6, 2, 3], [5, 4, 1, 6, 3, 2],

 [5, 4, 2, 1, 3, 6], [5, 4, 2, 1, 6, 3], [5, 4, 2, 3, 1, 6], [5, 4, 2, 3, 6, 1],

 [5, 4, 2, 6, 1, 3], [5, 4, 2, 6, 3, 1], [5, 4, 3, 1, 2, 6], [5, 4, 3, 1, 6, 2],

 [5, 4, 3, 2, 1, 6], [5, 4, 3, 2, 6, 1], [5, 4, 3, 6, 1, 2], [5, 4, 3, 6, 2, 1],

 [5, 4, 6, 1, 2, 3], [5, 4, 6, 1, 3, 2], [5, 4, 6, 2, 1, 3], [5, 4, 6, 2, 3, 1],

 [5, 4, 6, 3, 1, 2], [5, 4, 6, 3, 2, 1], [5, 6, 1, 2, 4, 3], [5, 6, 1, 3, 2, 4],

 [5, 6, 1, 3, 4, 2], [5, 6, 1, 4, 2, 3], [5, 6, 1, 4, 3, 2], [5, 6, 2, 1, 3, 4],

 [5, 6, 2, 1, 4, 3], [5, 6, 2, 3, 1, 4], [5, 6, 2, 3, 4, 1], [5, 6, 2, 4, 1, 3],

 [5, 6, 2, 4, 3, 1], [5, 6, 3, 1, 2, 4], [5, 6, 3, 1, 4, 2], [5, 6, 3, 2, 1, 4],

 [5, 6, 3, 2, 4, 1], [5, 6, 3, 4, 1, 2], [5, 6, 3, 4, 2, 1], [5, 6, 4, 1, 2, 3],

 [5, 6, 4, 1, 3, 2], [5, 6, 4, 2, 1, 3], [5, 6, 4, 2, 3, 1], [5, 6, 4, 3, 1, 2],

 [5, 6, 4, 3, 2, 1], [6, 1, 2, 3, 4, 5], [6, 1, 2, 3, 5, 4], [6, 1, 2, 4, 3, 5],

 [6, 1, 2, 4, 5, 3], [6, 1, 2, 5, 3, 4], [6, 1, 2, 5, 4, 3], [6, 1, 3, 2, 4, 5],

 [6, 1, 3, 2, 5, 4], [6, 1, 3, 4, 2, 5], [6, 1, 3, 4, 5, 2], [6, 1, 3, 5, 2, 4],

 [6, 1, 3, 5, 4, 2], [6, 1, 4, 2, 3, 5], [6, 1, 4, 2, 5, 3], [6, 1, 4, 3, 2, 5],

 [6, 1, 4, 3, 5, 2], [6, 1, 4, 5, 2, 3], [6, 1, 4, 5, 3, 2], [6, 1, 5, 2, 4, 3],

 [6, 1, 5, 3, 2, 4], [6, 1, 5, 3, 4, 2], [6, 1, 5, 4, 2, 3], [6, 1, 5, 4, 3, 2],

 [6, 2, 1, 3, 4, 5], [6, 2, 1, 3, 5, 4], [6, 2, 1, 4, 3, 5], [6, 2, 1, 4, 5, 3],

 [6, 2, 1, 5, 3, 4], [6, 2, 1, 5, 4, 3], [6, 2, 3, 1, 4, 5], [6, 2, 3, 1, 5, 4],

 [6, 2, 3, 4, 1, 5], [6, 2, 3, 4, 5, 1], [6, 2, 3, 5, 1, 4], [6, 2, 3, 5, 4, 1],

 [6, 2, 4, 1, 3, 5], [6, 2, 4, 1, 5, 3], [6, 2, 4, 3, 1, 5], [6, 2, 4, 3, 5, 1],

 [6, 2, 4, 5, 1, 3], [6, 2, 4, 5, 3, 1], [6, 2, 5, 1, 3, 4], [6, 2, 5, 1, 4, 3],

 [6, 2, 5, 3, 1, 4], [6, 2, 5, 3, 4, 1], [6, 2, 5, 4, 1, 3], [6, 2, 5, 4, 3, 1],

 [6, 3, 1, 2, 4, 5], [6, 3, 1, 2, 5, 4], [6, 3, 1, 4, 2, 5], [6, 3, 1, 4, 5, 2],

 [6, 3, 1, 5, 2, 4], [6, 3, 1, 5, 4, 2], [6, 3, 2, 1, 4, 5], [6, 3, 2, 1, 5, 4],

 [6, 3, 2, 4, 1, 5], [6, 3, 2, 4, 5, 1], [6, 3, 2, 5, 1, 4], [6, 3, 2, 5, 4, 1],

 [6, 3, 4, 1, 2, 5], [6, 3, 4, 1, 5, 2], [6, 3, 4, 2, 1, 5], [6, 3, 4, 2, 5, 1],

 [6, 3, 4, 5, 1, 2], [6, 3, 4, 5, 2, 1], [6, 3, 5, 1, 2, 4], [6, 3, 5, 1, 4, 2],

 [6, 3, 5, 2, 1, 4], [6, 3, 5, 2, 4, 1], [6, 3, 5, 4, 1, 2], [6, 3, 5, 4, 2, 1],

 [6, 4, 1, 2, 3, 5], [6, 4, 1, 2, 5, 3], [6, 4, 1, 3, 2, 5], [6, 4, 1, 3, 5, 2],

 [6, 4, 1, 5, 2, 3], [6, 4, 1, 5, 3, 2], [6, 4, 2, 1, 3, 5], [6, 4, 2, 1, 5, 3],

 [6, 4, 2, 3, 1, 5], [6, 4, 2, 3, 5, 1], [6, 4, 2, 5, 1, 3], [6, 4, 2, 5, 3, 1],

 [6, 4, 3, 1, 2, 5], [6, 4, 3, 1, 5, 2], [6, 4, 3, 2, 1, 5], [6, 4, 3, 2, 5, 1],

 [6, 4, 3, 5, 1, 2], [6, 4, 3, 5, 2, 1], [6, 4, 5, 1, 2, 3], [6, 4, 5, 1, 3, 2],

 [6, 4, 5, 2, 1, 3], [6, 4, 5, 2, 3, 1], [6, 4, 5, 3, 1, 2], [6, 4, 5, 3, 2, 1],

 [6, 5, 1, 2, 4, 3], [6, 5, 1, 3, 2, 4], [6, 5, 1, 3, 4, 2], [6, 5, 1, 4, 2, 3],

 [6, 5, 1, 4, 3, 2], [6, 5, 2, 1, 3, 4], [6, 5, 2, 1, 4, 3], [6, 5, 2, 3, 1, 4],

 [6, 5, 2, 3, 4, 1], [6, 5, 2, 4, 1, 3], [6, 5, 2, 4, 3, 1], [6, 5, 3, 1, 2, 4],

 [6, 5, 3, 1, 4, 2], [6, 5, 3, 2, 1, 4], [6, 5, 3, 2, 4, 1], [6, 5, 3, 4, 1, 2],

 [6, 5, 3, 4, 2, 1], [6, 5, 4, 1, 2, 3], [6, 5, 4, 1, 3, 2], [6, 5, 4, 2, 1, 3],

 [6, 5, 4, 2, 3, 1], [6, 5, 4, 3, 1, 2], [6, 5, 4, 3, 2, 1] ]

6. zadatak

Koliko ima različitih uređenih šestorki čije su komponente znamenke $0,1,2,\dotsc,9$ i koje imaju:

1. dvije jedinice, dvije dvojke i još dvije različite znamenke,
2. pet jednakih komponenti,
3. tri različita para jednakih komponenata,
4. simetrični raspored komponenti,
5. sve različite komponente,
6. sve različite komponente, ali u rastućem poretku?

Rješenje

a) dio

time len(list(filter(lambda x: x.count(1)==2 and x.count(2)==2 and len(set(x))==4, Tuples(range(10),6))))

CPU times: user 3.57 s, sys: 0 ns, total: 3.57 s
Wall time: 3.58 s

5040

b) dio

time len(list(filter(lambda x: x.count(x[0])==5 or x.count(x[1])==5, Tuples(range(10),6))))

CPU times: user 3.47 s, sys: 0 ns, total: 3.47 s
Wall time: 3.47 s

540

c) dio

%time len(list(filter(lambda x: x.count(x[0])==2 and x.count(x[1])==2 and x.count(x[2])==2\
                                and x.count(x[3])==2 and x.count(x[4])==2 and x.count(x[5])==2, Tuples(range(10),6))))

CPU times: user 3.2 s, sys: 6.15 ms, total: 3.2 s
Wall time: 3.21 s

10800

d) dio

time len(list(filter(lambda x: x[0]==x[-1] and x[1]==x[-2] and x[2]==x[-3], Tuples(range(10),6))))

CPU times: user 2.8 s, sys: 3.28 ms, total: 2.81 s
Wall time: 2.81 s

1000

e) dio

time len(list(filter(lambda x: len(set(x))==6, Tuples(range(10),6))))

CPU times: user 3.32 s, sys: 16.6 ms, total: 3.33 s
Wall time: 3.34 s

151200

time Arrangements(range(10),6).cardinality()

CPU times: user 426 ms, sys: 0 ns, total: 426 ms
Wall time: 429 ms

151200

f) dio

time len(list(filter(lambda x: len(set(x))==6 and x==sorted(x), Tuples(range(10),6))))

CPU times: user 3.22 s, sys: 3.25 ms, total: 3.23 s
Wall time: 3.25 s

210

%time
sve=Arrangements(range(10),6).list()
rast=list(filter(lambda x: list(x)==sorted(x),sve))
len(rast)

CPU times: user 5 µs, sys: 0 ns, total: 5 µs
Wall time: 9.3 µs

210

rast

[ [0, 1, 2, 3, 4, 5], [0, 1, 2, 3, 4, 6], [0, 1, 2, 3, 4, 7],

 [0, 1, 2, 3, 4, 8], [0, 1, 2, 3, 4, 9], [0, 1, 2, 3, 5, 6], [0, 1, 2, 3, 5, 7],

 [0, 1, 2, 3, 5, 8], [0, 1, 2, 3, 5, 9], [0, 1, 2, 3, 6, 7], [0, 1, 2, 3, 6, 8],

 [0, 1, 2, 3, 6, 9], [0, 1, 2, 3, 7, 8], [0, 1, 2, 3, 7, 9], [0, 1, 2, 3, 8, 9],

 [0, 1, 2, 4, 5, 6], [0, 1, 2, 4, 5, 7], [0, 1, 2, 4, 5, 8], [0, 1, 2, 4, 5, 9],

 [0, 1, 2, 4, 6, 7], [0, 1, 2, 4, 6, 8], [0, 1, 2, 4, 6, 9], [0, 1, 2, 4, 7, 8],

 [0, 1, 2, 4, 7, 9], [0, 1, 2, 4, 8, 9], [0, 1, 2, 5, 6, 7], [0, 1, 2, 5, 6, 8],

 [0, 1, 2, 5, 6, 9], [0, 1, 2, 5, 7, 8], [0, 1, 2, 5, 7, 9], [0, 1, 2, 5, 8, 9],

 [0, 1, 2, 6, 7, 8], [0, 1, 2, 6, 7, 9], [0, 1, 2, 6, 8, 9], [0, 1, 2, 7, 8, 9],

 [0, 1, 3, 4, 5, 6], [0, 1, 3, 4, 5, 7], [0, 1, 3, 4, 5, 8], [0, 1, 3, 4, 5, 9],

 [0, 1, 3, 4, 6, 7], [0, 1, 3, 4, 6, 8], [0, 1, 3, 4, 6, 9], [0, 1, 3, 4, 7, 8],

 [0, 1, 3, 4, 7, 9], [0, 1, 3, 4, 8, 9], [0, 1, 3, 5, 6, 7], [0, 1, 3, 5, 6, 8],

 [0, 1, 3, 5, 6, 9], [0, 1, 3, 5, 7, 8], [0, 1, 3, 5, 7, 9], [0, 1, 3, 5, 8, 9],

 [0, 1, 3, 6, 7, 8], [0, 1, 3, 6, 7, 9], [0, 1, 3, 6, 8, 9], [0, 1, 3, 7, 8, 9],

 [0, 1, 4, 5, 6, 7], [0, 1, 4, 5, 6, 8], [0, 1, 4, 5, 6, 9], [0, 1, 4, 5, 7, 8],

 [0, 1, 4, 5, 7, 9], [0, 1, 4, 5, 8, 9], [0, 1, 4, 6, 7, 8], [0, 1, 4, 6, 7, 9],

 [0, 1, 4, 6, 8, 9], [0, 1, 4, 7, 8, 9], [0, 1, 5, 6, 7, 8], [0, 1, 5, 6, 7, 9],

 [0, 1, 5, 6, 8, 9], [0, 1, 5, 7, 8, 9], [0, 1, 6, 7, 8, 9], [0, 2, 3, 4, 5, 6],

 [0, 2, 3, 4, 5, 7], [0, 2, 3, 4, 5, 8], [0, 2, 3, 4, 5, 9], [0, 2, 3, 4, 6, 7],

 [0, 2, 3, 4, 6, 8], [0, 2, 3, 4, 6, 9], [0, 2, 3, 4, 7, 8], [0, 2, 3, 4, 7, 9],

 [0, 2, 3, 4, 8, 9], [0, 2, 3, 5, 6, 7], [0, 2, 3, 5, 6, 8], [0, 2, 3, 5, 6, 9],

 [0, 2, 3, 5, 7, 8], [0, 2, 3, 5, 7, 9], [0, 2, 3, 5, 8, 9], [0, 2, 3, 6, 7, 8],

 [0, 2, 3, 6, 7, 9], [0, 2, 3, 6, 8, 9], [0, 2, 3, 7, 8, 9], [0, 2, 4, 5, 6, 7],

 [0, 2, 4, 5, 6, 8], [0, 2, 4, 5, 6, 9], [0, 2, 4, 5, 7, 8], [0, 2, 4, 5, 7, 9],

 [0, 2, 4, 5, 8, 9], [0, 2, 4, 6, 7, 8], [0, 2, 4, 6, 7, 9], [0, 2, 4, 6, 8, 9],

 [0, 2, 4, 7, 8, 9], [0, 2, 5, 6, 7, 8], [0, 2, 5, 6, 7, 9], [0, 2, 5, 6, 8, 9],

 [0, 2, 5, 7, 8, 9], [0, 2, 6, 7, 8, 9], [0, 3, 4, 5, 6, 7], [0, 3, 4, 5, 6, 8],

 [0, 3, 4, 5, 6, 9], [0, 3, 4, 5, 7, 8], [0, 3, 4, 5, 7, 9], [0, 3, 4, 5, 8, 9],

 [0, 3, 4, 6, 7, 8], [0, 3, 4, 6, 7, 9], [0, 3, 4, 6, 8, 9], [0, 3, 4, 7, 8, 9],

 [0, 3, 5, 6, 7, 8], [0, 3, 5, 6, 7, 9], [0, 3, 5, 6, 8, 9], [0, 3, 5, 7, 8, 9],

 [0, 3, 6, 7, 8, 9], [0, 4, 5, 6, 7, 8], [0, 4, 5, 6, 7, 9], [0, 4, 5, 6, 8, 9],

 [0, 4, 5, 7, 8, 9], [0, 4, 6, 7, 8, 9], [0, 5, 6, 7, 8, 9], [1, 2, 3, 4, 5, 6],

 [1, 2, 3, 4, 5, 7], [1, 2, 3, 4, 5, 8], [1, 2, 3, 4, 5, 9], [1, 2, 3, 4, 6, 7],

 [1, 2, 3, 4, 6, 8], [1, 2, 3, 4, 6, 9], [1, 2, 3, 4, 7, 8], [1, 2, 3, 4, 7, 9],

 [1, 2, 3, 4, 8, 9], [1, 2, 3, 5, 6, 7], [1, 2, 3, 5, 6, 8], [1, 2, 3, 5, 6, 9],

 [1, 2, 3, 5, 7, 8], [1, 2, 3, 5, 7, 9], [1, 2, 3, 5, 8, 9], [1, 2, 3, 6, 7, 8],

 [1, 2, 3, 6, 7, 9], [1, 2, 3, 6, 8, 9], [1, 2, 3, 7, 8, 9], [1, 2, 4, 5, 6, 7],

 [1, 2, 4, 5, 6, 8], [1, 2, 4, 5, 6, 9], [1, 2, 4, 5, 7, 8], [1, 2, 4, 5, 7, 9],

 [1, 2, 4, 5, 8, 9], [1, 2, 4, 6, 7, 8], [1, 2, 4, 6, 7, 9], [1, 2, 4, 6, 8, 9],

 [1, 2, 4, 7, 8, 9], [1, 2, 5, 6, 7, 8], [1, 2, 5, 6, 7, 9], [1, 2, 5, 6, 8, 9],

 [1, 2, 5, 7, 8, 9], [1, 2, 6, 7, 8, 9], [1, 3, 4, 5, 6, 7], [1, 3, 4, 5, 6, 8],

 [1, 3, 4, 5, 6, 9], [1, 3, 4, 5, 7, 8], [1, 3, 4, 5, 7, 9], [1, 3, 4, 5, 8, 9],

 [1, 3, 4, 6, 7, 8], [1, 3, 4, 6, 7, 9], [1, 3, 4, 6, 8, 9], [1, 3, 4, 7, 8, 9],

 [1, 3, 5, 6, 7, 8], [1, 3, 5, 6, 7, 9], [1, 3, 5, 6, 8, 9], [1, 3, 5, 7, 8, 9],

 [1, 3, 6, 7, 8, 9], [1, 4, 5, 6, 7, 8], [1, 4, 5, 6, 7, 9], [1, 4, 5, 6, 8, 9],

 [1, 4, 5, 7, 8, 9], [1, 4, 6, 7, 8, 9], [1, 5, 6, 7, 8, 9], [2, 3, 4, 5, 6, 7],

 [2, 3, 4, 5, 6, 8], [2, 3, 4, 5, 6, 9], [2, 3, 4, 5, 7, 8], [2, 3, 4, 5, 7, 9],

 [2, 3, 4, 5, 8, 9], [2, 3, 4, 6, 7, 8], [2, 3, 4, 6, 7, 9], [2, 3, 4, 6, 8, 9],

 [2, 3, 4, 7, 8, 9], [2, 3, 5, 6, 7, 8], [2, 3, 5, 6, 7, 9], [2, 3, 5, 6, 8, 9],

 [2, 3, 5, 7, 8, 9], [2, 3, 6, 7, 8, 9], [2, 4, 5, 6, 7, 8], [2, 4, 5, 6, 7, 9],

 [2, 4, 5, 6, 8, 9], [2, 4, 5, 7, 8, 9], [2, 4, 6, 7, 8, 9], [2, 5, 6, 7, 8, 9],

 [3, 4, 5, 6, 7, 8], [3, 4, 5, 6, 7, 9], [3, 4, 5, 6, 8, 9], [3, 4, 5, 7, 8, 9],

 [3, 4, 6, 7, 8, 9], [3, 5, 6, 7, 8, 9], [4, 5, 6, 7, 8, 9] ]

7. zadatak

Koliko ima nenegativnih cjelobrojnih rješenja jednadžba $x_1+x_2+x_3+x_4=13$?

Rješenje

time len(list(filter(lambda x: x[0]+x[1]+x[2]+x[3]==13, Tuples(range(14),4))))

CPU times: user 110 ms, sys: 12 µs, total: 111 ms
Wall time: 110 ms

560

list(filter(lambda x: x[0]+x[1]+x[2]+x[3]==13, Tuples(range(14),4)))

[ [ 13, 0, 0, 0 ], [ 12, 1, 0, 0 ], [ 11, 2, 0, 0 ], [ 10, 3, 0, 0 ],

 [ 9, 4, 0, 0 ], [ 8, 5, 0, 0 ], [ 7, 6, 0, 0 ], [ 6, 7, 0, 0 ], [ 5, 8, 0, 0 ],

 [ 4, 9, 0, 0 ], [ 3, 10, 0, 0 ], [ 2, 11, 0, 0 ], [ 1, 12, 0, 0 ],

 [ 0, 13, 0, 0 ], [ 12, 0, 1, 0 ], [ 11, 1, 1, 0 ], [ 10, 2, 1, 0 ],

 [ 9, 3, 1, 0 ], [ 8, 4, 1, 0 ], [ 7, 5, 1, 0 ], [ 6, 6, 1, 0 ], [ 5, 7, 1, 0 ],

 [ 4, 8, 1, 0 ], [ 3, 9, 1, 0 ], [ 2, 10, 1, 0 ], [ 1, 11, 1, 0 ],

 [ 0, 12, 1, 0 ], [ 11, 0, 2, 0 ], [ 10, 1, 2, 0 ], [ 9, 2, 2, 0 ],

 [ 8, 3, 2, 0 ], [ 7, 4, 2, 0 ], [ 6, 5, 2, 0 ], [ 5, 6, 2, 0 ], [ 4, 7, 2, 0 ],

 [ 3, 8, 2, 0 ], [ 2, 9, 2, 0 ], [ 1, 10, 2, 0 ], [ 0, 11, 2, 0 ],

 [ 10, 0, 3, 0 ], [ 9, 1, 3, 0 ], [ 8, 2, 3, 0 ], [ 7, 3, 3, 0 ],

 [ 6, 4, 3, 0 ], [ 5, 5, 3, 0 ], [ 4, 6, 3, 0 ], [ 3, 7, 3, 0 ], [ 2, 8, 3, 0 ],

 [ 1, 9, 3, 0 ], [ 0, 10, 3, 0 ], [ 9, 0, 4, 0 ], [ 8, 1, 4, 0 ],

 [ 7, 2, 4, 0 ], [ 6, 3, 4, 0 ], [ 5, 4, 4, 0 ], [ 4, 5, 4, 0 ], [ 3, 6, 4, 0 ],

 [ 2, 7, 4, 0 ], [ 1, 8, 4, 0 ], [ 0, 9, 4, 0 ], [ 8, 0, 5, 0 ], [ 7, 1, 5, 0 ],

 [ 6, 2, 5, 0 ], [ 5, 3, 5, 0 ], [ 4, 4, 5, 0 ], [ 3, 5, 5, 0 ], [ 2, 6, 5, 0 ],

 [ 1, 7, 5, 0 ], [ 0, 8, 5, 0 ], [ 7, 0, 6, 0 ], [ 6, 1, 6, 0 ], [ 5, 2, 6, 0 ],

 [ 4, 3, 6, 0 ], [ 3, 4, 6, 0 ], [ 2, 5, 6, 0 ], [ 1, 6, 6, 0 ], [ 0, 7, 6, 0 ],

 [ 6, 0, 7, 0 ], [ 5, 1, 7, 0 ], [ 4, 2, 7, 0 ], [ 3, 3, 7, 0 ], [ 2, 4, 7, 0 ],

 [ 1, 5, 7, 0 ], [ 0, 6, 7, 0 ], [ 5, 0, 8, 0 ], [ 4, 1, 8, 0 ], [ 3, 2, 8, 0 ],

 [ 2, 3, 8, 0 ], [ 1, 4, 8, 0 ], [ 0, 5, 8, 0 ], [ 4, 0, 9, 0 ], [ 3, 1, 9, 0 ],

 [ 2, 2, 9, 0 ], [ 1, 3, 9, 0 ], [ 0, 4, 9, 0 ], [ 3, 0, 10, 0 ],

 [ 2, 1, 10, 0 ], [ 1, 2, 10, 0 ], [ 0, 3, 10, 0 ], [ 2, 0, 11, 0 ],

 [ 1, 1, 11, 0 ], [ 0, 2, 11, 0 ], [ 1, 0, 12, 0 ], [ 0, 1, 12, 0 ],

 [ 0, 0, 13, 0 ], [ 12, 0, 0, 1 ], [ 11, 1, 0, 1 ], [ 10, 2, 0, 1 ],

 [ 9, 3, 0, 1 ], [ 8, 4, 0, 1 ], [ 7, 5, 0, 1 ], [ 6, 6, 0, 1 ], [ 5, 7, 0, 1 ],

 [ 4, 8, 0, 1 ], [ 3, 9, 0, 1 ], [ 2, 10, 0, 1 ], [ 1, 11, 0, 1 ],

 [ 0, 12, 0, 1 ], [ 11, 0, 1, 1 ], [ 10, 1, 1, 1 ], [ 9, 2, 1, 1 ],

 [ 8, 3, 1, 1 ], [ 7, 4, 1, 1 ], [ 6, 5, 1, 1 ], [ 5, 6, 1, 1 ], [ 4, 7, 1, 1 ],

 [ 3, 8, 1, 1 ], [ 2, 9, 1, 1 ], [ 1, 10, 1, 1 ], [ 0, 11, 1, 1 ],

 [ 10, 0, 2, 1 ], [ 9, 1, 2, 1 ], [ 8, 2, 2, 1 ], [ 7, 3, 2, 1 ],

 [ 6, 4, 2, 1 ], [ 5, 5, 2, 1 ], [ 4, 6, 2, 1 ], [ 3, 7, 2, 1 ], [ 2, 8, 2, 1 ],

 [ 1, 9, 2, 1 ], [ 0, 10, 2, 1 ], [ 9, 0, 3, 1 ], [ 8, 1, 3, 1 ],

 [ 7, 2, 3, 1 ], [ 6, 3, 3, 1 ], [ 5, 4, 3, 1 ], [ 4, 5, 3, 1 ], [ 3, 6, 3, 1 ],

 [ 2, 7, 3, 1 ], [ 1, 8, 3, 1 ], [ 0, 9, 3, 1 ], [ 8, 0, 4, 1 ], [ 7, 1, 4, 1 ],

 [ 6, 2, 4, 1 ], [ 5, 3, 4, 1 ], [ 4, 4, 4, 1 ], [ 3, 5, 4, 1 ], [ 2, 6, 4, 1 ],

 [ 1, 7, 4, 1 ], [ 0, 8, 4, 1 ], [ 7, 0, 5, 1 ], [ 6, 1, 5, 1 ], [ 5, 2, 5, 1 ],

 [ 4, 3, 5, 1 ], [ 3, 4, 5, 1 ], [ 2, 5, 5, 1 ], [ 1, 6, 5, 1 ], [ 0, 7, 5, 1 ],

 [ 6, 0, 6, 1 ], [ 5, 1, 6, 1 ], [ 4, 2, 6, 1 ], [ 3, 3, 6, 1 ], [ 2, 4, 6, 1 ],

 [ 1, 5, 6, 1 ], [ 0, 6, 6, 1 ], [ 5, 0, 7, 1 ], [ 4, 1, 7, 1 ], [ 3, 2, 7, 1 ],

 [ 2, 3, 7, 1 ], [ 1, 4, 7, 1 ], [ 0, 5, 7, 1 ], [ 4, 0, 8, 1 ], [ 3, 1, 8, 1 ],

 [ 2, 2, 8, 1 ], [ 1, 3, 8, 1 ], [ 0, 4, 8, 1 ], [ 3, 0, 9, 1 ], [ 2, 1, 9, 1 ],

 [ 1, 2, 9, 1 ], [ 0, 3, 9, 1 ], [ 2, 0, 10, 1 ], [ 1, 1, 10, 1 ],

 [ 0, 2, 10, 1 ], [ 1, 0, 11, 1 ], [ 0, 1, 11, 1 ], [ 0, 0, 12, 1 ],

 [ 11, 0, 0, 2 ], [ 10, 1, 0, 2 ], [ 9, 2, 0, 2 ], [ 8, 3, 0, 2 ],

 [ 7, 4, 0, 2 ], [ 6, 5, 0, 2 ], [ 5, 6, 0, 2 ], [ 4, 7, 0, 2 ], [ 3, 8, 0, 2 ],

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 [ 4, 6, 1, 2 ], [ 3, 7, 1, 2 ], [ 2, 8, 1, 2 ], [ 1, 9, 1, 2 ],

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 [ 1, 8, 2, 2 ], [ 0, 9, 2, 2 ], [ 8, 0, 3, 2 ], [ 7, 1, 3, 2 ], [ 6, 2, 3, 2 ],

 [ 5, 3, 3, 2 ], [ 4, 4, 3, 2 ], [ 3, 5, 3, 2 ], [ 2, 6, 3, 2 ], [ 1, 7, 3, 2 ],

 [ 0, 8, 3, 2 ], [ 7, 0, 4, 2 ], [ 6, 1, 4, 2 ], [ 5, 2, 4, 2 ], [ 4, 3, 4, 2 ],

 [ 3, 4, 4, 2 ], [ 2, 5, 4, 2 ], [ 1, 6, 4, 2 ], [ 0, 7, 4, 2 ], [ 6, 0, 5, 2 ],

 [ 5, 1, 5, 2 ], [ 4, 2, 5, 2 ], [ 3, 3, 5, 2 ], [ 2, 4, 5, 2 ], [ 1, 5, 5, 2 ],

 [ 0, 6, 5, 2 ], [ 5, 0, 6, 2 ], [ 4, 1, 6, 2 ], [ 3, 2, 6, 2 ], [ 2, 3, 6, 2 ],

 [ 1, 4, 6, 2 ], [ 0, 5, 6, 2 ], [ 4, 0, 7, 2 ], [ 3, 1, 7, 2 ], [ 2, 2, 7, 2 ],

 [ 1, 3, 7, 2 ], [ 0, 4, 7, 2 ], [ 3, 0, 8, 2 ], [ 2, 1, 8, 2 ], [ 1, 2, 8, 2 ],

 [ 0, 3, 8, 2 ], [ 2, 0, 9, 2 ], [ 1, 1, 9, 2 ], [ 0, 2, 9, 2 ],

 [ 1, 0, 10, 2 ], [ 0, 1, 10, 2 ], [ 0, 0, 11, 2 ], [ 10, 0, 0, 3 ],

 [ 9, 1, 0, 3 ], [ 8, 2, 0, 3 ], [ 7, 3, 0, 3 ], [ 6, 4, 0, 3 ], [ 5, 5, 0, 3 ],

 [ 4, 6, 0, 3 ], [ 3, 7, 0, 3 ], [ 2, 8, 0, 3 ], [ 1, 9, 0, 3 ],

 [ 0, 10, 0, 3 ], [ 9, 0, 1, 3 ], [ 8, 1, 1, 3 ], [ 7, 2, 1, 3 ],

 [ 6, 3, 1, 3 ], [ 5, 4, 1, 3 ], [ 4, 5, 1, 3 ], [ 3, 6, 1, 3 ], [ 2, 7, 1, 3 ],

 [ 1, 8, 1, 3 ], [ 0, 9, 1, 3 ], [ 8, 0, 2, 3 ], [ 7, 1, 2, 3 ], [ 6, 2, 2, 3 ],

 [ 5, 3, 2, 3 ], [ 4, 4, 2, 3 ], [ 3, 5, 2, 3 ], [ 2, 6, 2, 3 ], [ 1, 7, 2, 3 ],

 [ 0, 8, 2, 3 ], [ 7, 0, 3, 3 ], [ 6, 1, 3, 3 ], [ 5, 2, 3, 3 ], [ 4, 3, 3, 3 ],

 [ 3, 4, 3, 3 ], [ 2, 5, 3, 3 ], [ 1, 6, 3, 3 ], [ 0, 7, 3, 3 ], [ 6, 0, 4, 3 ],

 [ 5, 1, 4, 3 ], [ 4, 2, 4, 3 ], [ 3, 3, 4, 3 ], [ 2, 4, 4, 3 ], [ 1, 5, 4, 3 ],

 [ 0, 6, 4, 3 ], [ 5, 0, 5, 3 ], [ 4, 1, 5, 3 ], [ 3, 2, 5, 3 ], [ 2, 3, 5, 3 ],

 [ 1, 4, 5, 3 ], [ 0, 5, 5, 3 ], [ 4, 0, 6, 3 ], [ 3, 1, 6, 3 ], [ 2, 2, 6, 3 ],

 [ 1, 3, 6, 3 ], [ 0, 4, 6, 3 ], [ 3, 0, 7, 3 ], [ 2, 1, 7, 3 ], [ 1, 2, 7, 3 ],

 [ 0, 3, 7, 3 ], [ 2, 0, 8, 3 ], [ 1, 1, 8, 3 ], [ 0, 2, 8, 3 ], [ 1, 0, 9, 3 ],

 [ 0, 1, 9, 3 ], [ 0, 0, 10, 3 ], [ 9, 0, 0, 4 ], [ 8, 1, 0, 4 ],

 [ 7, 2, 0, 4 ], [ 6, 3, 0, 4 ], [ 5, 4, 0, 4 ], [ 4, 5, 0, 4 ], [ 3, 6, 0, 4 ],

 [ 2, 7, 0, 4 ], [ 1, 8, 0, 4 ], [ 0, 9, 0, 4 ], [ 8, 0, 1, 4 ], [ 7, 1, 1, 4 ],

 [ 6, 2, 1, 4 ], [ 5, 3, 1, 4 ], [ 4, 4, 1, 4 ], [ 3, 5, 1, 4 ], [ 2, 6, 1, 4 ],

 [ 1, 7, 1, 4 ], [ 0, 8, 1, 4 ], [ 7, 0, 2, 4 ], [ 6, 1, 2, 4 ], [ 5, 2, 2, 4 ],

 [ 4, 3, 2, 4 ], [ 3, 4, 2, 4 ], [ 2, 5, 2, 4 ], [ 1, 6, 2, 4 ], [ 0, 7, 2, 4 ],

 [ 6, 0, 3, 4 ], [ 5, 1, 3, 4 ], [ 4, 2, 3, 4 ], [ 3, 3, 3, 4 ], [ 2, 4, 3, 4 ],

 [ 1, 5, 3, 4 ], [ 0, 6, 3, 4 ], [ 5, 0, 4, 4 ], [ 4, 1, 4, 4 ], [ 3, 2, 4, 4 ],

 [ 2, 3, 4, 4 ], [ 1, 4, 4, 4 ], [ 0, 5, 4, 4 ], [ 4, 0, 5, 4 ], [ 3, 1, 5, 4 ],

 [ 2, 2, 5, 4 ], [ 1, 3, 5, 4 ], [ 0, 4, 5, 4 ], [ 3, 0, 6, 4 ], [ 2, 1, 6, 4 ],

 [ 1, 2, 6, 4 ], [ 0, 3, 6, 4 ], [ 2, 0, 7, 4 ], [ 1, 1, 7, 4 ], [ 0, 2, 7, 4 ],

 [ 1, 0, 8, 4 ], [ 0, 1, 8, 4 ], [ 0, 0, 9, 4 ], [ 8, 0, 0, 5 ], [ 7, 1, 0, 5 ],

 [ 6, 2, 0, 5 ], [ 5, 3, 0, 5 ], [ 4, 4, 0, 5 ], [ 3, 5, 0, 5 ], [ 2, 6, 0, 5 ],

 [ 1, 7, 0, 5 ], [ 0, 8, 0, 5 ], [ 7, 0, 1, 5 ], [ 6, 1, 1, 5 ], [ 5, 2, 1, 5 ],

 [ 4, 3, 1, 5 ], [ 3, 4, 1, 5 ], [ 2, 5, 1, 5 ], [ 1, 6, 1, 5 ], [ 0, 7, 1, 5 ],

 [ 6, 0, 2, 5 ], [ 5, 1, 2, 5 ], [ 4, 2, 2, 5 ], [ 3, 3, 2, 5 ], [ 2, 4, 2, 5 ],

 [ 1, 5, 2, 5 ], [ 0, 6, 2, 5 ], [ 5, 0, 3, 5 ], [ 4, 1, 3, 5 ], [ 3, 2, 3, 5 ],

 [ 2, 3, 3, 5 ], [ 1, 4, 3, 5 ], [ 0, 5, 3, 5 ], [ 4, 0, 4, 5 ], [ 3, 1, 4, 5 ],

 [ 2, 2, 4, 5 ], [ 1, 3, 4, 5 ], [ 0, 4, 4, 5 ], [ 3, 0, 5, 5 ], [ 2, 1, 5, 5 ],

 [ 1, 2, 5, 5 ], [ 0, 3, 5, 5 ], [ 2, 0, 6, 5 ], [ 1, 1, 6, 5 ], [ 0, 2, 6, 5 ],

 [ 1, 0, 7, 5 ], [ 0, 1, 7, 5 ], [ 0, 0, 8, 5 ], [ 7, 0, 0, 6 ], [ 6, 1, 0, 6 ],

 [ 5, 2, 0, 6 ], [ 4, 3, 0, 6 ], [ 3, 4, 0, 6 ], [ 2, 5, 0, 6 ], [ 1, 6, 0, 6 ],

 [ 0, 7, 0, 6 ], [ 6, 0, 1, 6 ], [ 5, 1, 1, 6 ], [ 4, 2, 1, 6 ], [ 3, 3, 1, 6 ],

 [ 2, 4, 1, 6 ], [ 1, 5, 1, 6 ], [ 0, 6, 1, 6 ], [ 5, 0, 2, 6 ], [ 4, 1, 2, 6 ],

 [ 3, 2, 2, 6 ], [ 2, 3, 2, 6 ], [ 1, 4, 2, 6 ], [ 0, 5, 2, 6 ], [ 4, 0, 3, 6 ],

 [ 3, 1, 3, 6 ], [ 2, 2, 3, 6 ], [ 1, 3, 3, 6 ], [ 0, 4, 3, 6 ], [ 3, 0, 4, 6 ],

 [ 2, 1, 4, 6 ], [ 1, 2, 4, 6 ], [ 0, 3, 4, 6 ], [ 2, 0, 5, 6 ], [ 1, 1, 5, 6 ],

 [ 0, 2, 5, 6 ], [ 1, 0, 6, 6 ], [ 0, 1, 6, 6 ], [ 0, 0, 7, 6 ], [ 6, 0, 0, 7 ],

 [ 5, 1, 0, 7 ], [ 4, 2, 0, 7 ], [ 3, 3, 0, 7 ], [ 2, 4, 0, 7 ], [ 1, 5, 0, 7 ],

 [ 0, 6, 0, 7 ], [ 5, 0, 1, 7 ], [ 4, 1, 1, 7 ], [ 3, 2, 1, 7 ], [ 2, 3, 1, 7 ],

 [ 1, 4, 1, 7 ], [ 0, 5, 1, 7 ], [ 4, 0, 2, 7 ], [ 3, 1, 2, 7 ], [ 2, 2, 2, 7 ],

 [ 1, 3, 2, 7 ], [ 0, 4, 2, 7 ], [ 3, 0, 3, 7 ], [ 2, 1, 3, 7 ], [ 1, 2, 3, 7 ],

 [ 0, 3, 3, 7 ], [ 2, 0, 4, 7 ], [ 1, 1, 4, 7 ], [ 0, 2, 4, 7 ], [ 1, 0, 5, 7 ],

 [ 0, 1, 5, 7 ], [ 0, 0, 6, 7 ], [ 5, 0, 0, 8 ], [ 4, 1, 0, 8 ], [ 3, 2, 0, 8 ],

 [ 2, 3, 0, 8 ], [ 1, 4, 0, 8 ], [ 0, 5, 0, 8 ], [ 4, 0, 1, 8 ], [ 3, 1, 1, 8 ],

 [ 2, 2, 1, 8 ], [ 1, 3, 1, 8 ], [ 0, 4, 1, 8 ], [ 3, 0, 2, 8 ], [ 2, 1, 2, 8 ],

 [ 1, 2, 2, 8 ], [ 0, 3, 2, 8 ], [ 2, 0, 3, 8 ], [ 1, 1, 3, 8 ], [ 0, 2, 3, 8 ],

 [ 1, 0, 4, 8 ], [ 0, 1, 4, 8 ], [ 0, 0, 5, 8 ], [ 4, 0, 0, 9 ], [ 3, 1, 0, 9 ],

 [ 2, 2, 0, 9 ], [ 1, 3, 0, 9 ], [ 0, 4, 0, 9 ], [ 3, 0, 1, 9 ], [ 2, 1, 1, 9 ],

 [ 1, 2, 1, 9 ], [ 0, 3, 1, 9 ], [ 2, 0, 2, 9 ], [ 1, 1, 2, 9 ], [ 0, 2, 2, 9 ],

 [ 1, 0, 3, 9 ], [ 0, 1, 3, 9 ], [ 0, 0, 4, 9 ], [ 3, 0, 0, 10 ],

 [ 2, 1, 0, 10 ], [ 1, 2, 0, 10 ], [ 0, 3, 0, 10 ], [ 2, 0, 1, 10 ],

 [ 1, 1, 1, 10 ], [ 0, 2, 1, 10 ], [ 1, 0, 2, 10 ], [ 0, 1, 2, 10 ],

 [ 0, 0, 3, 10 ], [ 2, 0, 0, 11 ], [ 1, 1, 0, 11 ], [ 0, 2, 0, 11 ],

 [ 1, 0, 1, 11 ], [ 0, 1, 1, 11 ], [ 0, 0, 2, 11 ], [ 1, 0, 0, 12 ],

 [ 0, 1, 0, 12 ], [ 0, 0, 1, 12 ], [ 0, 0, 0, 13 ] ]

8. zadatak

Koliko rješenja u skupu $\mathbb{N}$ ima jednadžba $x_1+x_2+x_3+x_4=13$?

Rješenje

time len(list(filter(lambda x: x[0]+x[1]+x[2]+x[3]==13, Tuples(range(1,14),4))))

CPU times: user 87.1 ms, sys: 0 ns, total: 87.1 ms
Wall time: 87.7 ms

220

list(filter(lambda x: x[0]+x[1]+x[2]+x[3]==13, Tuples(range(1,14),4)))

[ [ 10, 1, 1, 1 ], [ 9, 2, 1, 1 ], [ 8, 3, 1, 1 ], [ 7, 4, 1, 1 ],

 [ 6, 5, 1, 1 ], [ 5, 6, 1, 1 ], [ 4, 7, 1, 1 ], [ 3, 8, 1, 1 ], [ 2, 9, 1, 1 ],

 [ 1, 10, 1, 1 ], [ 9, 1, 2, 1 ], [ 8, 2, 2, 1 ], [ 7, 3, 2, 1 ],

 [ 6, 4, 2, 1 ], [ 5, 5, 2, 1 ], [ 4, 6, 2, 1 ], [ 3, 7, 2, 1 ], [ 2, 8, 2, 1 ],

 [ 1, 9, 2, 1 ], [ 8, 1, 3, 1 ], [ 7, 2, 3, 1 ], [ 6, 3, 3, 1 ], [ 5, 4, 3, 1 ],

 [ 4, 5, 3, 1 ], [ 3, 6, 3, 1 ], [ 2, 7, 3, 1 ], [ 1, 8, 3, 1 ], [ 7, 1, 4, 1 ],

 [ 6, 2, 4, 1 ], [ 5, 3, 4, 1 ], [ 4, 4, 4, 1 ], [ 3, 5, 4, 1 ], [ 2, 6, 4, 1 ],

 [ 1, 7, 4, 1 ], [ 6, 1, 5, 1 ], [ 5, 2, 5, 1 ], [ 4, 3, 5, 1 ], [ 3, 4, 5, 1 ],

 [ 2, 5, 5, 1 ], [ 1, 6, 5, 1 ], [ 5, 1, 6, 1 ], [ 4, 2, 6, 1 ], [ 3, 3, 6, 1 ],

 [ 2, 4, 6, 1 ], [ 1, 5, 6, 1 ], [ 4, 1, 7, 1 ], [ 3, 2, 7, 1 ], [ 2, 3, 7, 1 ],

 [ 1, 4, 7, 1 ], [ 3, 1, 8, 1 ], [ 2, 2, 8, 1 ], [ 1, 3, 8, 1 ], [ 2, 1, 9, 1 ],

 [ 1, 2, 9, 1 ], [ 1, 1, 10, 1 ], [ 9, 1, 1, 2 ], [ 8, 2, 1, 2 ],

 [ 7, 3, 1, 2 ], [ 6, 4, 1, 2 ], [ 5, 5, 1, 2 ], [ 4, 6, 1, 2 ], [ 3, 7, 1, 2 ],

 [ 2, 8, 1, 2 ], [ 1, 9, 1, 2 ], [ 8, 1, 2, 2 ], [ 7, 2, 2, 2 ], [ 6, 3, 2, 2 ],

 [ 5, 4, 2, 2 ], [ 4, 5, 2, 2 ], [ 3, 6, 2, 2 ], [ 2, 7, 2, 2 ], [ 1, 8, 2, 2 ],

 [ 7, 1, 3, 2 ], [ 6, 2, 3, 2 ], [ 5, 3, 3, 2 ], [ 4, 4, 3, 2 ], [ 3, 5, 3, 2 ],

 [ 2, 6, 3, 2 ], [ 1, 7, 3, 2 ], [ 6, 1, 4, 2 ], [ 5, 2, 4, 2 ], [ 4, 3, 4, 2 ],

 [ 3, 4, 4, 2 ], [ 2, 5, 4, 2 ], [ 1, 6, 4, 2 ], [ 5, 1, 5, 2 ], [ 4, 2, 5, 2 ],

 [ 3, 3, 5, 2 ], [ 2, 4, 5, 2 ], [ 1, 5, 5, 2 ], [ 4, 1, 6, 2 ], [ 3, 2, 6, 2 ],

 [ 2, 3, 6, 2 ], [ 1, 4, 6, 2 ], [ 3, 1, 7, 2 ], [ 2, 2, 7, 2 ], [ 1, 3, 7, 2 ],

 [ 2, 1, 8, 2 ], [ 1, 2, 8, 2 ], [ 1, 1, 9, 2 ], [ 8, 1, 1, 3 ], [ 7, 2, 1, 3 ],

 [ 6, 3, 1, 3 ], [ 5, 4, 1, 3 ], [ 4, 5, 1, 3 ], [ 3, 6, 1, 3 ], [ 2, 7, 1, 3 ],

 [ 1, 8, 1, 3 ], [ 7, 1, 2, 3 ], [ 6, 2, 2, 3 ], [ 5, 3, 2, 3 ], [ 4, 4, 2, 3 ],

 [ 3, 5, 2, 3 ], [ 2, 6, 2, 3 ], [ 1, 7, 2, 3 ], [ 6, 1, 3, 3 ], [ 5, 2, 3, 3 ],

 [ 4, 3, 3, 3 ], [ 3, 4, 3, 3 ], [ 2, 5, 3, 3 ], [ 1, 6, 3, 3 ], [ 5, 1, 4, 3 ],

 [ 4, 2, 4, 3 ], [ 3, 3, 4, 3 ], [ 2, 4, 4, 3 ], [ 1, 5, 4, 3 ], [ 4, 1, 5, 3 ],

 [ 3, 2, 5, 3 ], [ 2, 3, 5, 3 ], [ 1, 4, 5, 3 ], [ 3, 1, 6, 3 ], [ 2, 2, 6, 3 ],

 [ 1, 3, 6, 3 ], [ 2, 1, 7, 3 ], [ 1, 2, 7, 3 ], [ 1, 1, 8, 3 ], [ 7, 1, 1, 4 ],

 [ 6, 2, 1, 4 ], [ 5, 3, 1, 4 ], [ 4, 4, 1, 4 ], [ 3, 5, 1, 4 ], [ 2, 6, 1, 4 ],

 [ 1, 7, 1, 4 ], [ 6, 1, 2, 4 ], [ 5, 2, 2, 4 ], [ 4, 3, 2, 4 ], [ 3, 4, 2, 4 ],

 [ 2, 5, 2, 4 ], [ 1, 6, 2, 4 ], [ 5, 1, 3, 4 ], [ 4, 2, 3, 4 ], [ 3, 3, 3, 4 ],

 [ 2, 4, 3, 4 ], [ 1, 5, 3, 4 ], [ 4, 1, 4, 4 ], [ 3, 2, 4, 4 ], [ 2, 3, 4, 4 ],

 [ 1, 4, 4, 4 ], [ 3, 1, 5, 4 ], [ 2, 2, 5, 4 ], [ 1, 3, 5, 4 ], [ 2, 1, 6, 4 ],

 [ 1, 2, 6, 4 ], [ 1, 1, 7, 4 ], [ 6, 1, 1, 5 ], [ 5, 2, 1, 5 ], [ 4, 3, 1, 5 ],

 [ 3, 4, 1, 5 ], [ 2, 5, 1, 5 ], [ 1, 6, 1, 5 ], [ 5, 1, 2, 5 ], [ 4, 2, 2, 5 ],

 [ 3, 3, 2, 5 ], [ 2, 4, 2, 5 ], [ 1, 5, 2, 5 ], [ 4, 1, 3, 5 ], [ 3, 2, 3, 5 ],

 [ 2, 3, 3, 5 ], [ 1, 4, 3, 5 ], [ 3, 1, 4, 5 ], [ 2, 2, 4, 5 ], [ 1, 3, 4, 5 ],

 [ 2, 1, 5, 5 ], [ 1, 2, 5, 5 ], [ 1, 1, 6, 5 ], [ 5, 1, 1, 6 ], [ 4, 2, 1, 6 ],

 [ 3, 3, 1, 6 ], [ 2, 4, 1, 6 ], [ 1, 5, 1, 6 ], [ 4, 1, 2, 6 ], [ 3, 2, 2, 6 ],

 [ 2, 3, 2, 6 ], [ 1, 4, 2, 6 ], [ 3, 1, 3, 6 ], [ 2, 2, 3, 6 ], [ 1, 3, 3, 6 ],

 [ 2, 1, 4, 6 ], [ 1, 2, 4, 6 ], [ 1, 1, 5, 6 ], [ 4, 1, 1, 7 ], [ 3, 2, 1, 7 ],

 [ 2, 3, 1, 7 ], [ 1, 4, 1, 7 ], [ 3, 1, 2, 7 ], [ 2, 2, 2, 7 ], [ 1, 3, 2, 7 ],

 [ 2, 1, 3, 7 ], [ 1, 2, 3, 7 ], [ 1, 1, 4, 7 ], [ 3, 1, 1, 8 ], [ 2, 2, 1, 8 ],

 [ 1, 3, 1, 8 ], [ 2, 1, 2, 8 ], [ 1, 2, 2, 8 ], [ 1, 1, 3, 8 ], [ 2, 1, 1, 9 ],

 [ 1, 2, 1, 9 ], [ 1, 1, 2, 9 ], [ 1, 1, 1, 10 ] ]

9. zadatak

Koliko ima nenegativnih cjelobrojnih rješenja jednadžba $x_1+x_2+x_3+x_4=13$ pri čemu je $x_1\geq8$?

Rješenje

time len(list(filter(lambda x: x[0]+x[1]+x[2]+x[3]==13 and x[0]>=8, Tuples(range(14),4))))

CPU times: user 107 ms, sys: 3.23 ms, total: 110 ms
Wall time: 110 ms

56

list(filter(lambda x: x[0]+x[1]+x[2]+x[3]==13 and x[0]>=8, Tuples(range(14),4)))

[ [ 13, 0, 0, 0 ], [ 12, 1, 0, 0 ], [ 11, 2, 0, 0 ], [ 10, 3, 0, 0 ],

 [ 9, 4, 0, 0 ], [ 8, 5, 0, 0 ], [ 12, 0, 1, 0 ], [ 11, 1, 1, 0 ],

 [ 10, 2, 1, 0 ], [ 9, 3, 1, 0 ], [ 8, 4, 1, 0 ], [ 11, 0, 2, 0 ],

 [ 10, 1, 2, 0 ], [ 9, 2, 2, 0 ], [ 8, 3, 2, 0 ], [ 10, 0, 3, 0 ],

 [ 9, 1, 3, 0 ], [ 8, 2, 3, 0 ], [ 9, 0, 4, 0 ], [ 8, 1, 4, 0 ], [ 8, 0, 5, 0 ],

 [ 12, 0, 0, 1 ], [ 11, 1, 0, 1 ], [ 10, 2, 0, 1 ], [ 9, 3, 0, 1 ],

 [ 8, 4, 0, 1 ], [ 11, 0, 1, 1 ], [ 10, 1, 1, 1 ], [ 9, 2, 1, 1 ],

 [ 8, 3, 1, 1 ], [ 10, 0, 2, 1 ], [ 9, 1, 2, 1 ], [ 8, 2, 2, 1 ],

 [ 9, 0, 3, 1 ], [ 8, 1, 3, 1 ], [ 8, 0, 4, 1 ], [ 11, 0, 0, 2 ],

 [ 10, 1, 0, 2 ], [ 9, 2, 0, 2 ], [ 8, 3, 0, 2 ], [ 10, 0, 1, 2 ],

 [ 9, 1, 1, 2 ], [ 8, 2, 1, 2 ], [ 9, 0, 2, 2 ], [ 8, 1, 2, 2 ], [ 8, 0, 3, 2 ],

 [ 10, 0, 0, 3 ], [ 9, 1, 0, 3 ], [ 8, 2, 0, 3 ], [ 9, 0, 1, 3 ],

 [ 8, 1, 1, 3 ], [ 8, 0, 2, 3 ], [ 9, 0, 0, 4 ], [ 8, 1, 0, 4 ], [ 8, 0, 1, 4 ],

 [ 8, 0, 0, 5 ] ]

10. zadatak

Koliko ima prirodnih brojeva manjih od 1000 koji nisu djeljivi sa 7?

Rješenje

len(list(filter(lambda x: x%7!=0, range(1,1000))))

857

list(filter(lambda x: x%7!=0, range(1,1000)))

[ 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 23, 24,

 25, 26, 27, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47,

 48, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 71,

 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 90, 92, 93, 94,

 95, 96, 97, 99, 100, 101, 102, 103, 104, 106, 107, 108, 109, 110, 111, 113,

 114, 115, 116, 117, 118, 120, 121, 122, 123, 124, 125, 127, 128, 129, 130, 131,

 132, 134, 135, 136, 137, 138, 139, 141, 142, 143, 144, 145, 146, 148, 149, 150,

 151, 152, 153, 155, 156, 157, 158, 159, 160, 162, 163, 164, 165, 166, 167, 169,

 170, 171, 172, 173, 174, 176, 177, 178, 179, 180, 181, 183, 184, 185, 186, 187,

 188, 190, 191, 192, 193, 194, 195, 197, 198, 199, 200, 201, 202, 204, 205, 206,

 207, 208, 209, 211, 212, 213, 214, 215, 216, 218, 219, 220, 221, 222, 223, 225,

 226, 227, 228, 229, 230, 232, 233, 234, 235, 236, 237, 239, 240, 241, 242, 243,

 244, 246, 247, 248, 249, 250, 251, 253, 254, 255, 256, 257, 258, 260, 261, 262,

 263, 264, 265, 267, 268, 269, 270, 271, 272, 274, 275, 276, 277, 278, 279, 281,

 282, 283, 284, 285, 286, 288, 289, 290, 291, 292, 293, 295, 296, 297, 298, 299,

 300, 302, 303, 304, 305, 306, 307, 309, 310, 311, 312, 313, 314, 316, 317, 318,

 319, 320, 321, 323, 324, 325, 326, 327, 328, 330, 331, 332, 333, 334, 335, 337,

 338, 339, 340, 341, 342, 344, 345, 346, 347, 348, 349, 351, 352, 353, 354, 355,

 356, 358, 359, 360, 361, 362, 363, 365, 366, 367, 368, 369, 370, 372, 373, 374,

 375, 376, 377, 379, 380, 381, 382, 383, 384, 386, 387, 388, 389, 390, 391, 393,

 394, 395, 396, 397, 398, 400, 401, 402, 403, 404, 405, 407, 408, 409, 410, 411,

 412, 414, 415, 416, 417, 418, 419, 421, 422, 423, 424, 425, 426, 428, 429, 430,

 431, 432, 433, 435, 436, 437, 438, 439, 440, 442, 443, 444, 445, 446, 447, 449,

 450, 451, 452, 453, 454, 456, 457, 458, 459, 460, 461, 463, 464, 465, 466, 467,

 468, 470, 471, 472, 473, 474, 475, 477, 478, 479, 480, 481, 482, 484, 485, 486,

 487, 488, 489, 491, 492, 493, 494, 495, 496, 498, 499, 500, 501, 502, 503, 505,

 506, 507, 508, 509, 510, 512, 513, 514, 515, 516, 517, 519, 520, 521, 522, 523,

 524, 526, 527, 528, 529, 530, 531, 533, 534, 535, 536, 537, 538, 540, 541, 542,

 543, 544, 545, 547, 548, 549, 550, 551, 552, 554, 555, 556, 557, 558, 559, 561,

 562, 563, 564, 565, 566, 568, 569, 570, 571, 572, 573, 575, 576, 577, 578, 579,

 580, 582, 583, 584, 585, 586, 587, 589, 590, 591, 592, 593, 594, 596, 597, 598,

 599, 600, 601, 603, 604, 605, 606, 607, 608, 610, 611, 612, 613, 614, 615, 617,

 618, 619, 620, 621, 622, 624, 625, 626, 627, 628, 629, 631, 632, 633, 634, 635,

 636, 638, 639, 640, 641, 642, 643, 645, 646, 647, 648, 649, 650, 652, 653, 654,

 655, 656, 657, 659, 660, 661, 662, 663, 664, 666, 667, 668, 669, 670, 671, 673,

 674, 675, 676, 677, 678, 680, 681, 682, 683, 684, 685, 687, 688, 689, 690, 691,

 692, 694, 695, 696, 697, 698, 699, 701, 702, 703, 704, 705, 706, 708, 709, 710,

 711, 712, 713, 715, 716, 717, 718, 719, 720, 722, 723, 724, 725, 726, 727, 729,

 730, 731, 732, 733, 734, 736, 737, 738, 739, 740, 741, 743, 744, 745, 746, 747,

 748, 750, 751, 752, 753, 754, 755, 757, 758, 759, 760, 761, 762, 764, 765, 766,

 767, 768, 769, 771, 772, 773, 774, 775, 776, 778, 779, 780, 781, 782, 783, 785,

 786, 787, 788, 789, 790, 792, 793, 794, 795, 796, 797, 799, 800, 801, 802, 803,

 804, 806, 807, 808, 809, 810, 811, 813, 814, 815, 816, 817, 818, 820, 821, 822,

 823, 824, 825, 827, 828, 829, 830, 831, 832, 834, 835, 836, 837, 838, 839, 841,

 842, 843, 844, 845, 846, 848, 849, 850, 851, 852, 853, 855, 856, 857, 858, 859,

 860, 862, 863, 864, 865, 866, 867, 869, 870, 871, 872, 873, 874, 876, 877, 878,

 879, 880, 881, 883, 884, 885, 886, 887, 888, 890, 891, 892, 893, 894, 895, 897,

 898, 899, 900, 901, 902, 904, 905, 906, 907, 908, 909, 911, 912, 913, 914, 915,

 916, 918, 919, 920, 921, 922, 923, 925, 926, 927, 928, 929, 930, 932, 933, 934,

 935, 936, 937, 939, 940, 941, 942, 943, 944, 946, 947, 948, 949, 950, 951, 953,

 954, 955, 956, 957, 958, 960, 961, 962, 963, 964, 965, 967, 968, 969, 970, 971,

 972, 974, 975, 976, 977, 978, 979, 981, 982, 983, 984, 985, 986, 988, 989, 990,

 991, 992, 993, 995, 996, 997, 998, 999 ]

11. zadatak

Izračunajte broj svih peteroznamenkastih brojeva koji:

1. sadrže barem jednu devetku,
2. sadrže barem jednu devetku ili barem jednu osmicu,
3. sadrže barem jednu devetku i barem jednu osmicu.

Rješenje

a) dio

len(list(filter(lambda x: x[0]!=0 and x.count(9)>0, Tuples(range(10),5))))

37512

b) dio

len(list(filter(lambda x: x[0]!=0 and (x.count(9)>0 or x.count(8)>0), Tuples(range(10),5))))

61328

c) dio

len(list(filter(lambda x: x[0]!=0 and x.count(9)>0 and x.count(8)>0, Tuples(range(10),5))))

13696

12. zadatak

Koliko ima brojeva od 1 do 1000 koji su djeljivi s 3, a nisu djeljivi niti s jednim od brojeva 2, 5 i 7?

Rješenje

len(list(filter(lambda x: x%3==0 and x%2!=0 and x%5!=0 and x%7!=0, range(1,1001))))

115

list(filter(lambda x: x%3==0 and x%2!=0 and x%5!=0 and x%7!=0, range(1,1001)))

[ 3, 9, 27, 33, 39, 51, 57, 69, 81, 87, 93, 99, 111, 117, 123, 129, 141, 153,

 159, 171, 177, 183, 201, 207, 213, 219, 237, 243, 249, 261, 267, 279, 291, 297,

 303, 309, 321, 327, 333, 339, 351, 363, 369, 381, 387, 393, 411, 417, 423, 429,

 447, 453, 459, 471, 477, 489, 501, 507, 513, 519, 531, 537, 543, 549, 561, 573,

 579, 591, 597, 603, 621, 627, 633, 639, 657, 663, 669, 681, 687, 699, 711, 717,

 723, 729, 741, 747, 753, 759, 771, 783, 789, 801, 807, 813, 831, 837, 843, 849,

 867, 873, 879, 891, 897, 909, 921, 927, 933, 939, 951, 957, 963, 969, 981, 993,

 999 ]

Rekurzivne relacije

maxima.load('solve_rec')

"/usr/share/maxima/5.44.0/share/solve_rec/solve_rec.mac"

%display latex

Zadatak

Niz $(a_n)$ zadan je rekurzivno s $a_1=\frac{3}{2},\ a_n=5a_{n-1}-1,\ n\geq2$. Napišite prvih 50 članova niza i pronađite formulu za opći član niza.

Rješenje

def niz(n):
    xn=3/2
    i=1
    while i<=n:
        yield xn
        xn=5*xn-1
        i+=1

list(niz(4))

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left[\frac{3}{2}, \frac{13}{2}, \frac{63}{2}, \frac{313}{2}\right]$$

maxima('solve_rec(a[n]=5*a[n-1]-1, a[n],a[1]=3/2)')

$$\newcommand{\Bold}[1]{\mathbf{#1}}a_{n}={{5^{n}}\over{4}}+{{1}\over{2^2}}$$

%%maxima
solve_rec(a[n]=5*a[n-1]-1, a[n],a[1]=3/2)

a[n]=5^n/4+1/2^2

Zadatak

Riješite sljedeće rekurzivne relacije:

1. $a_{n+2}=4a_{n+1}-3a_n$,   $a_1=1, a_2=7$
2. $F_n=F_{n-1}+F_{n-2}$,   $F_0=0, F_1=1$
3. $a_{n+3}=a_{n}$,   $a_0=0, a_1=0, a_2=1$

Rješenje

a) dio

maxima('solve_rec(a[n+2]=4*a[n+1]-3*a[n], a[n],a[1]=1,a[2]=7)')

$$\newcommand{\Bold}[1]{\mathbf{#1}}a_{n}=3^{n}-2$$

b) dio

maxima('solve_rec(F[n]=F[n-1]+F[n-2], F[n],F[0]=0,F[1]=1)')

$$\newcommand{\Bold}[1]{\mathbf{#1}}F_{n}={{\left({{\sqrt{5}+1}\over{2}}\right)^{n}}\over{\sqrt{5}}}-{{ \left(-{{\sqrt{5}-1}\over{2}}\right)^{n}}\over{\sqrt{5}}}$$

c) dio

maxima('solve_rec(a[n+3]=a[n], a[n],a[0]=0,a[1]=0,a[2]=1)')

$$\newcommand{\Bold}[1]{\mathbf{#1}}a_{n}=-{{\left(-{{\sqrt{3}\,i+1}\over{2}}\right)^{n}\,\left(\sqrt{3 }\,i+1\right)}\over{6}}+{{\left({{\sqrt{3}\,i-1}\over{2}}\right)^{n} \,\left(\sqrt{3}\,i-1\right)}\over{6}}+{{1}\over{3}}$$

Zadatak

Riješite rekurziju $a_{n+4}=8a_{n+3}-18a_{n+2}+27a_n$ uz početne uvjete $a_1=0$, $a_2=-1$, $a_3=2$, $a_4=5$.

Rješenje

maxima('solve_rec(a[n+4]=8*a[n+3]-18*a[n+2]+27*a[n], a[n],a[1]=0,a[2]=-1,a[3]=2,a[4]=5)')

$$\newcommand{\Bold}[1]{\mathbf{#1}}a_{n}=\left(-{{n^2}\over{27}}+{{5\,n}\over{18}}-{{97}\over{216}} \right)\,3^{n}-{{5\,\left(-1\right)^{n}}\over{8}}$$

Zadatak

Riješite sljedeće rekurzivne relacije:

1. $2a_{n+2}-a_{n+1}-a_n=2^n,\quad a_1=0, a_2=-1$
2. $a_{n+1}=5a_n+4n^2+2n+5,\quad a_1=2$
3. $2a_{n+2}-5a_{n+1}+2a_n=10,\quad a_0=6, a_1=-2$
4. $a_{n+2}-4a_{n+1}+4a_n=2^n,\quad a_1=0, a_2=1$
5. $a_{n+2}=4a_{n+1}-4a_n+n^2\cdot3^n,\quad a_1=0, a_2=1$
6. $a_{n+2}-4a_{n+1}+4a_n=3^n-2^n,\quad a_0=1, a_1=3$

Rješenje

a) dio

maxima('solve_rec(2*a[n+2]-a[n+1]-a[n]=2^n, a[n],a[1]=0,a[2]=-1)')

$$\newcommand{\Bold}[1]{\mathbf{#1}}a_{n}={{2^{n}}\over{5}}-{{7\,2^{2-n}\,\left(-1\right)^{n}}\over{15 }}-{{4}\over{3}}$$

b) dio

maxima('solve_rec(a[n+1]=5*a[n]+4*n^2+2*n+5, a[n],a[1]=2)')

$$\newcommand{\Bold}[1]{\mathbf{#1}}a_{n}={{23\,5^{n-1}}\over{4}}-{{4\,n^2+4\,n+7}\over{4}}$$

c) dio

maxima('solve_rec(2*a[n+2]-5*a[n+1]+2*a[n]=10, a[n],a[0]=6,a[1]=-2)')

$$\newcommand{\Bold}[1]{\mathbf{#1}}a_{n}=2^{4-n}-10$$

d) dio

maxima('solve_rec(a[n+2]-4*a[n+1]+4*a[n]=2^n, a[n],a[1]=0,a[2]=1)')

$$\newcommand{\Bold}[1]{\mathbf{#1}}a_{n}=n^2\,2^{n-3}-n\,2^{n-3}$$

e) dio

maxima('solve_rec(a[n+2]=4*a[n+1]-4*a[n]+n^2*3^n, a[n],a[1]=0,a[2]=1)')

$$\newcommand{\Bold}[1]{\mathbf{#1}}a_{n}=-4\,n\,3^{n+1}+16\,3^{n+1}+n^2\,3^{n}+\left(-{{29\,n}\over{4 }}-{{193}\over{4}}\right)\,2^{n}$$

f) dio

maxima('solve_rec(a[n+2]-4*a[n+1]+4*a[n]=3^n-2^n, a[n],a[0]=1,a[1]=3)')

$$\newcommand{\Bold}[1]{\mathbf{#1}}a_{n}=3^{n}-n^2\,2^{n-3}+n\,2^{n-3}$$

Zadatak

Riješite rekurziju $a_n=n^2a_{n-1}$ uz početni uvjet $a_0=1$.

Rješenje

maxima('solve_rec(a[n]=n^2*a[n-1], a[n],a[0]=1)')

$$\newcommand{\Bold}[1]{\mathbf{#1}}a_{n}=n!^2$$

Zadatak

Riješite sljedeće rekurzivne relacije:

1. $a_{n+1}=\frac{a_n-2}{a_n+4},\quad a_0=0$
2. $a_{n+1}=1+\frac{1}{a_n},\quad a_0=1$
3. $a_{n+1}=a_n-a_na_{n+1},\quad a_0=1$

Rješenje

a) dio

maxima('solve_rec(a[n+1]=(a[n]-2)/(a[n]+4), a[n],a[0]=0)')

$$\newcommand{\Bold}[1]{\mathbf{#1}}a_{n}={{2^{n+1}-2\,3^{n+1}}\over{2^{n}-2\,3^{n}}}-4$$

b) dio

maxima('solve_rec(a[n+1]=1+1/a[n], a[n],a[0]=1)')

$$\newcommand{\Bold}[1]{\mathbf{#1}}a_{n}={{\left({{\sqrt{5}+1}\over{2}}\right)^{n+1}-{{\left(\sqrt{5}- 3\right)\,\left(-{{\sqrt{5}-1}\over{2}}\right)^{n+1}}\over{2}} }\over{\left({{\sqrt{5}+1}\over{2}}\right)^{n}-{{\left(\sqrt{5}-3 \right)\,\left(-{{\sqrt{5}-1}\over{2}}\right)^{n}}\over{2}}}}$$

c) dio

maxima('solve_rec(a[n+1]=a[n]-a[n]*a[n+1], a[n],a[0]=1)')

$$\newcommand{\Bold}[1]{\mathbf{#1}}a_{n}={{n+2}\over{n+1}}-1$$

Funkcije izvodnice

Zadatak

Na koliko je načina moguće razmjestiti 20 jednakih kuglica u 5 različitih kutija tako da u svakoj kutiji budu barem dvije, a najviše sedam kuglica?

Rješenje

taylor((x^2+x^3+x^4+x^5+x^6+x^7)^5,x,0,22)

$$\newcommand{\Bold}[1]{\mathbf{#1}}780 \, x^{22} + 735 \, x^{21} + 651 \, x^{20} + 540 \, x^{19} + 420 \, x^{18} + 305 \, x^{17} + 205 \, x^{16} + 126 \, x^{15} + 70 \, x^{14} + 35 \, x^{13} + 15 \, x^{12} + 5 \, x^{11} + x^{10}$$

taylor((x^2+x^3+x^4+x^5+x^6+x^7)^5,x,0,22).coefficient(x^20)

$$\newcommand{\Bold}[1]{\mathbf{#1}}651$$

Zadatak

Koliko rješenja ima jednadžba $x_1+x_2+x_3+x_4=13$, gdje su svi $x_1, x_2, x_3, x_4$ nenegativni cijeli brojevi i $x_1\geq 8$ ?

Rješenje

Možemo raditi s formalnim redovima potencija nad $\mathbb{Z}$

R.<x>=PowerSeriesRing(ZZ)
f1=sum([x^n for n in range(8,14)])+O(x^14)
f2=sum([x^n for n in range(14)])+O(x^14)

f1

$$\newcommand{\Bold}[1]{\mathbf{#1}}x^{8} + x^{9} + x^{10} + x^{11} + x^{12} + x^{13} + O(x^{14})$$

f2

$$\newcommand{\Bold}[1]{\mathbf{#1}}1 + x + x^{2} + x^{3} + x^{4} + x^{5} + x^{6} + x^{7} + x^{8} + x^{9} + x^{10} + x^{11} + x^{12} + x^{13} + O(x^{14})$$

f2^3

$$\newcommand{\Bold}[1]{\mathbf{#1}}1 + 3x + 6x^{2} + 10x^{3} + 15x^{4} + 21x^{5} + 28x^{6} + 36x^{7} + 45x^{8} + 55x^{9} + 66x^{10} + 78x^{11} + 91x^{12} + 105x^{13} + O(x^{14})$$

f1*f2^3

$$\newcommand{\Bold}[1]{\mathbf{#1}}x^{8} + 4x^{9} + 10x^{10} + 20x^{11} + 35x^{12} + 56x^{13} + O(x^{14})$$

(f1*f2^3).coefficients()

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left[1, 4, 10, 20, 35, 56\right]$$

(f1*f2^3).padded_list(14)

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left[0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 10, 20, 35, 56\right]$$

(f1*f2^3).padded_list(14)[-1]

$$\newcommand{\Bold}[1]{\mathbf{#1}}56$$

ili možemo raditi samo s polinomima

var('t')
P.<t>=PolynomialRing(ZZ)
p1=sum([t^n for n in range(8,14)])
p2=sum([t^n for n in range(14)])

p1

$$\newcommand{\Bold}[1]{\mathbf{#1}}t^{13} + t^{12} + t^{11} + t^{10} + t^{9} + t^{8}$$

p2

$$\newcommand{\Bold}[1]{\mathbf{#1}}t^{13} + t^{12} + t^{11} + t^{10} + t^{9} + t^{8} + t^{7} + t^{6} + t^{5} + t^{4} + t^{3} + t^{2} + t + 1$$

pp=p1*p2^3;pp

$$\newcommand{\Bold}[1]{\mathbf{#1}}t^{52} + 4t^{51} + 10t^{50} + 20t^{49} + 35t^{48} + 56t^{47} + 83t^{46} + 116t^{45} + 155t^{44} + 200t^{43} + 251t^{42} + 308t^{41} + 371t^{40} + 440t^{39} + 512t^{38} + 584t^{37} + 653t^{36} + 716t^{35} + 770t^{34} + 812t^{33} + 842t^{32} + 860t^{31} + 866t^{30} + 860t^{29} + 842t^{28} + 812t^{27} + 770t^{26} + 716t^{25} + 653t^{24} + 584t^{23} + 512t^{22} + 440t^{21} + 371t^{20} + 308t^{19} + 251t^{18} + 200t^{17} + 155t^{16} + 116t^{15} + 83t^{14} + 56t^{13} + 35t^{12} + 20t^{11} + 10t^{10} + 4t^{9} + t^{8}$$

pp.coefficients()

$$\newcommand{\Bold}[1]{\mathbf{#1}}\left[1, 4, 10, 20, 35, 56, 83, 116, 155, 200, 251, 308, 371, 440, 512, 584, 653, 716, 770, 812, 842, 860, 866, 860, 842, 812, 770, 716, 653, 584, 512, 440, 371, 308, 251, 200, 155, 116, 83, 56, 35, 20, 10, 4, 1\right]$$

pp.padded_list()[13]

$$\newcommand{\Bold}[1]{\mathbf{#1}}56$$

ili možemo raditi direktno sa simboličkim izrazima bez da prethodno definiramo algebarsku strukturu

var('y')
f1=sum([y^n for n in range(8,14)])
f2=sum([y^n for n in range(14)])

f1

$$\newcommand{\Bold}[1]{\mathbf{#1}}y^{13} + y^{12} + y^{11} + y^{10} + y^{9} + y^{8}$$

f2

$$\newcommand{\Bold}[1]{\mathbf{#1}}y^{13} + y^{12} + y^{11} + y^{10} + y^{9} + y^{8} + y^{7} + y^{6} + y^{5} + y^{4} + y^{3} + y^{2} + y + 1$$

f1*f2^3

$$\newcommand{\Bold}[1]{\mathbf{#1}}{\left(y^{13} + y^{12} + y^{11} + y^{10} + y^{9} + y^{8} + y^{7} + y^{6} + y^{5} + y^{4} + y^{3} + y^{2} + y + 1\right)}^{3} {\left(y^{13} + y^{12} + y^{11} + y^{10} + y^{9} + y^{8}\right)}$$

expand(f1*f2^3)

$$\newcommand{\Bold}[1]{\mathbf{#1}}y^{52} + 4 \, y^{51} + 10 \, y^{50} + 20 \, y^{49} + 35 \, y^{48} + 56 \, y^{47} + 83 \, y^{46} + 116 \, y^{45} + 155 \, y^{44} + 200 \, y^{43} + 251 \, y^{42} + 308 \, y^{41} + 371 \, y^{40} + 440 \, y^{39} + 512 \, y^{38} + 584 \, y^{37} + 653 \, y^{36} + 716 \, y^{35} + 770 \, y^{34} + 812 \, y^{33} + 842 \, y^{32} + 860 \, y^{31} + 866 \, y^{30} + 860 \, y^{29} + 842 \, y^{28} + 812 \, y^{27} + 770 \, y^{26} + 716 \, y^{25} + 653 \, y^{24} + 584 \, y^{23} + 512 \, y^{22} + 440 \, y^{21} + 371 \, y^{20} + 308 \, y^{19} + 251 \, y^{18} + 200 \, y^{17} + 155 \, y^{16} + 116 \, y^{15} + 83 \, y^{14} + 56 \, y^{13} + 35 \, y^{12} + 20 \, y^{11} + 10 \, y^{10} + 4 \, y^{9} + y^{8}$$

expand(f1*f2^3).coefficient(y^13)

$$\newcommand{\Bold}[1]{\mathbf{#1}}56$$

Zadatak

Odredite funkciju izvodnicu za broj načina da se postigne suma $n$ pri bacanju deset različitih igraćih kocki. Izvedite iz te funkcije izvodnice funkciju izvodnicu za broj načina da se postigne parna suma, te izračunajte broj različitih načina dobivanja parne sume.

Rješenje

funkcija izvodnica za bacanje 10 igraćih kocki

f(y)=(y+y^2+y^3+y^4+y^5+y^6)^10

funkcija izvodnica za parne sume

p(y)=(f(y)+f(-y))/2

ukupni broj načina dobivanja parne sume

p(1)

$$\newcommand{\Bold}[1]{\mathbf{#1}}30233088$$

Zadatak

Na koliko je načina moguće podijeliti 18 različitih igrački između troje djece tako da svako dijete dobije barem jednu igračku?

Rješenje

Možemo raditi s formalnim redovima potencija nad $\mathbb{Q}$

R.<x>=PowerSeriesRing(QQ)
f=sum([x^n for n in range(1,19)])+O(x^19)

f

$$\newcommand{\Bold}[1]{\mathbf{#1}}x + x^{2} + x^{3} + x^{4} + x^{5} + x^{6} + x^{7} + x^{8} + x^{9} + x^{10} + x^{11} + x^{12} + x^{13} + x^{14} + x^{15} + x^{16} + x^{17} + x^{18} + O(x^{19})$$

f.ogf_to_egf()

$$\newcommand{\Bold}[1]{\mathbf{#1}}x + \frac{1}{2}x^{2} + \frac{1}{6}x^{3} + \frac{1}{24}x^{4} + \frac{1}{120}x^{5} + \frac{1}{720}x^{6} + \frac{1}{5040}x^{7} + \frac{1}{40320}x^{8} + \frac{1}{362880}x^{9} + \frac{1}{3628800}x^{10} + \frac{1}{39916800}x^{11} + \frac{1}{479001600}x^{12} + \frac{1}{6227020800}x^{13} + \frac{1}{87178291200}x^{14} + \frac{1}{1307674368000}x^{15} + \frac{1}{20922789888000}x^{16} + \frac{1}{355687428096000}x^{17} + \frac{1}{6402373705728000}x^{18}$$

f1=f.ogf_to_egf()^3;f1

$$\newcommand{\Bold}[1]{\mathbf{#1}}x^{3} + \frac{3}{2}x^{4} + \frac{5}{4}x^{5} + \frac{3}{4}x^{6} + \frac{43}{120}x^{7} + \frac{23}{160}x^{8} + \frac{605}{12096}x^{9} + \frac{311}{20160}x^{10} + \frac{2591}{604800}x^{11} + \frac{437}{403200}x^{12} + \frac{115}{456192}x^{13} + \frac{289}{5322240}x^{14} + \frac{2375101}{217945728000}x^{15} + \frac{1190281}{581188608000}x^{16} + \frac{50489}{139485265920}x^{17} + \frac{14039}{232475443200}x^{18} + \frac{132227}{13857951744000}x^{19} + \frac{727577}{508124897280000}x^{20} + \frac{18340457}{89633231880192000}x^{21} + \frac{8338237}{298777439600640000}x^{22} + \frac{717907}{197193110136422400}x^{23} + \frac{142453}{313004936724480000}x^{24} + \frac{12923717}{236631732163706880000}x^{25} + \frac{1656833}{262924146848563200000}x^{26} + \frac{58147249}{83057737989461114880000}x^{27} + \frac{145324967}{1938013886420759347200000}x^{28} + \frac{2861897}{369145502175382732800000}x^{29} + \frac{2770993}{3577871790315248025600000}x^{30} + \frac{1450039}{19380138864207593472000000}x^{31} + \frac{5205821}{744197332385571589324800000}x^{32} + \frac{17279953}{27287235520804291608576000000}x^{33} + \frac{27275041}{491170239374477248954368000000}x^{34} + \frac{91918187}{19483086161854264208523264000000}x^{35} + \frac{817171447}{2104173305480260534520512512000000}x^{36} + \frac{65166071}{2104173305480260534520512512000000}x^{37} + \frac{3355241}{1402782203653507023013675008000000}x^{38} + \frac{146736523}{820627589137301608462999879680000000}x^{39} + \frac{141476411}{10941701188497354779506665062400000000}x^{40} + \frac{29657099}{32825103565492064338519995187200000000}x^{41} + \frac{9332593}{153183816638962966913093310873600000000}x^{42} + \frac{5456447}{1378654349750666702217839797862400000000}x^{43} + \frac{454313}{1838205799667555602957119730483200000000}x^{44} + \frac{3671281}{248157782955120006399211163615232000000000}x^{45} + \frac{419801}{496315565910240012798422327230464000000000}x^{46} + \frac{22819}{496315565910240012798422327230464000000000}x^{47} + \frac{56257}{23823147163691520614324271707062272000000000}x^{48} + \frac{2707}{23823147163691520614324271707062272000000000}x^{49} + \frac{241}{47646294327383041228648543414124544000000000}x^{50} + \frac{31}{151872563168533443916317232132521984000000000}x^{51} + \frac{1}{138854914896944863009204326521162956800000000}x^{52} + \frac{1}{4859922021393070205322151428240703488000000000}x^{53} + \frac{1}{262435789155225791087396177124997988352000000000}x^{54}$$

f1.padded_list()[18]*factorial(18)

$$\newcommand{\Bold}[1]{\mathbf{#1}}386634060$$

ili možemo raditi samo s polinomima

P.<t>=PolynomialRing(QQ)
p1=sum([(t^n)/factorial(n) for n in range(1,19)])

(p1^3).coefficients(sparse=False)[18]*factorial(18)

$$\newcommand{\Bold}[1]{\mathbf{#1}}386634060$$

ili možemo raditi direktno sa simboličkim izrazima bez da prethodno definiramo algebarsku strukturu

f1=sum([(y^n)/factorial(n) for n in range(1,19)])

expand(f1^3).coefficient(y^18)*factorial(18)

$$\newcommand{\Bold}[1]{\mathbf{#1}}386634060$$

Zadatak

Na koliko je načina moguće podijeliti 18 različitih igrački između troje djece tako da svako dijete dobije barem dvije igračke, ali ne više od 6 igrački?

Rješenje

f2=sum([(y^n)/factorial(n) for n in range(2,7)])

f2

$$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{720} \, y^{6} + \frac{1}{120} \, y^{5} + \frac{1}{24} \, y^{4} + \frac{1}{6} \, y^{3} + \frac{1}{2} \, y^{2}$$

expand(f2^3).coefficient(y^18)*factorial(18)

$$\newcommand{\Bold}[1]{\mathbf{#1}}17153136$$