Derangement
Permutation of the elements of a set in which no element appears in its original position / From Wikipedia, the free encyclopedia
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In combinatorial mathematics, a derangement is a permutation of the elements of a set in which no element appears in its original position. In other words, a derangement is a permutation that has no fixed points.
Table of values | |||
---|---|---|---|
Permutations, | Derangements, | ||
0 | 1 | 1 | = 1 |
1 | 1 | 0 | = 0 |
2 | 2 | 1 | = 0.5 |
3 | 6 | 2 | ≈0.33333 | 33333
4 | 24 | 9 | = 0.375 |
5 | 120 | 44 | ≈0.36666 | 66667
6 | 720 | 265 | ≈0.36805 | 55556
7 | 5,040 | 1,854 | ≈0.36785,71429 |
8 | 40,320 | 14,833 | ≈0.36788 | 19444
9 | 362,880 | 133,496 | ≈0.36787 | 91887
10 | 3,628,800 | 1,334,961 | ≈0.36787 | 94643
11 | 39,916,800 | 14,684,570 | ≈0.36787 | 94392
12 | 479,001,600 | 176,214,841 | ≈0.36787 | 94413
13 | 6,227,020,800 | 2,290,792,932 | ≈0.36787 | 94412
14 | 87,178,291,200 | 32,071,101,049 | ≈0.36787 | 94412
15 | 1,307,674,368,000 | 481,066,515,734 | ≈0.36787 | 94412
16 | 20,922,789,888,000 | 7,697,064,251,745 | ≈0.36787 | 94412
17 | 355,687,428,096,000 | 130,850,092,279,664 | ≈0.36787 | 94412
18 | 6,402,373,705,728,000 | 2,355,301,661,033,953 | ≈0.36787 | 94412
19 | 121,645,100,408,832,000 | 44,750,731,559,645,106 | ≈0.36787 | 94412
20 | 2,432,902,008,176,640,000 | 895,014,631,192,902,121 | ≈0.36787 | 94412
21 | 51,090,942,171,709,440,000 | 18,795,307,255,050,944,540 | ≈0.36787 | 94412
22 | 1,124,000,727,777,607,680,000 | 413,496,759,611,120,779,881 | ≈0.36787 | 94412
23 | 25,852,016,738,884,976,640,000 | 9,510,425,471,055,777,937,262 | ≈0.36787 | 94412
24 | 620,448,401,733,239,439,360,000 | 228,250,211,305,338,670,494,289 | ≈0.36787 | 94412
25 | 15,511,210,043,330,985,984,000,000 | 5,706,255,282,633,466,762,357,224 | ≈0.36787 | 94412
26 | 403,291,461,126,605,635,584,000,000 | 148,362,637,348,470,135,821,287,825 | ≈0.36787 | 94412
27 | 10,888,869,450,418,352,160,768,000,000 | 4,005,791,208,408,693,667,174,771,274 | ≈0.36787 | 94412
28 | 304,888,344,611,713,860,501,504,000,000 | 112,162,153,835,443,422,680,893,595,673 | ≈0.36787 | 94412
29 | 8,841,761,993,739,701,954,543,616,000,000 | 3,252,702,461,227,859,257,745,914,274,516 | ≈0.36787 | 94412
30 | 265,252,859,812,191,058,636,308,480,000,000 | 97,581,073,836,835,777,732,377,428,235,481 | ≈0.36787 | 94412
The number of derangements of a set of size n is known as the subfactorial of n or the n-th derangement number or n-th de Montmort number (after Pierre Remond de Montmort). Notations for subfactorials in common use include !n, Dn, dn, or n¡.[1][2]
For n > 0, the subfactorial !n equals the nearest integer to n!/e, where n! denotes the factorial of n and e is Euler's number.[3]
The problem of counting derangements was first considered by Pierre Raymond de Montmort in his Essay d'analyse sur les jeux de hazard.[4] in 1708; he solved it in 1713, as did Nicholas Bernoulli at about the same time.