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==Computational complexity== It is [[NP-complete]] to determine whether a given [[permutation group]] (described by a given set of permutations that generate it) contains any derangements.<ref> {{cite journal | last = Lubiw | first = Anna | author-link = Anna Lubiw | year = 1981 | title = Some NP-complete problems similar to graph isomorphism | journal = [[SIAM Journal on Computing]] | volume = 10 | issue = 1 | pages = 11β21 | doi = 10.1137/0210002 | mr = 605600 }} </ref><ref> {{cite book | last = Babai | first = LΓ‘szlΓ³ | author-link = LΓ‘szlΓ³ Babai | year = 1995 | contribution = Automorphism groups, isomorphism, reconstruction | title = Handbook of Combinatorics | volume = 1, 2 | at = ch. 27, pp. 1447β1540 | location = Amsterdam, NL | publisher = Elsevier | mr = 1373683 | url = http://people.cs.uchicago.edu/~laci/handbook/handbookchapter27.pdf | via = cs.uchicago.edu }} </ref> : {| class="wikitable collapsible collapsed" style="margin:0; width:100%" |+ Table of factorial and derangement values<span id="derangements_and_factorials_0_30" class="anchor"></span> |- ! scope="col" | <math>n</math> ! scope="col" class="nowrap" | Permutations, <math>n!</math> ! scope="col" class="nowrap" | Derangements, <math>!n</math> ! scope="col" | <math>\frac{!n}{n!}</math> |- | style="text-align: center" | 0 | 1 <span style="font-size:80%; float:right;">=1×10<sup>0</sup></span> | 1 <span style="font-size:80%; float:right;">=1×10<sup>0</sup></span> | = 1 |- | style="text-align: center" | 1 | 1 <span style="font-size:80%; float:right;">=1×10<sup>0</sup></span> | 0 | = 0 |- | style="text-align: center" | 2 | 2 <span style="font-size:80%; float:right;">=2×10<sup>0</sup></span> | 1 <span style="font-size:80%; float:right;">=1×10<sup>0</sup></span> | = 0.5 |- | style="text-align: center" | 3 | 6 <span style="font-size:80%; float:right;">=6×10<sup>0</sup></span> | 2 <span style="font-size:80%; float:right;">=2×10<sup>0</sup></span> |align="right"| β0.33333 33333 |- | style="text-align: center" | 4 | 24 <span style="font-size:80%; float:right;">=2.4×10<sup>1</sup></span> | 9 <span style="font-size:80%; float:right;">=9×10<sup>0</sup></span> | = 0.375 |-style="border-top:2px solid #aaaaaa;" | style="text-align: center" | 5 | 120 <span style="font-size:80%; float:right;">=1.20×10<sup>2</sup></span> | 44 <span style="font-size:80%; float:right;">=4.4×10<sup>1</sup></span> |align="right"| β0.36666 66667 |- | style="text-align: center" | 6 | 720 <span style="font-size:80%; float:right;">=7.20×10<sup>2</sup></span> | 265 <span style="font-size:80%; float:right;">=2.65×10<sup>2</sup></span> |align="right"| β0.36805 55556 |- | style="text-align: center" | 7 | 5,040 <span style="font-size:80%; float:right;">=5.04×10<sup>3</sup></span> | 1,854 <span style="font-size:80%; float:right;">≈1.85×10<sup>3</sup></span> |align="right"| β0.36785,71429 |- | style="text-align: center" | 8 | 40,320 <span style="font-size:80%; float:right;">≈4.03×10<sup>4</sup></span> | 14,833 <span style="font-size:80%; float:right;">≈1.48×10<sup>4</sup></span> |align="right"| β0.36788 19444 |- | style="text-align: center" | 9 | 362,880 <span style="font-size:80%; float:right;">≈3.63×10<sup>5</sup></span> | 133,496 <span style="font-size:80%; float:right;">≈1.33×10<sup>5</sup></span> |align="right"| β0.36787 91887 |-style="border-top:2px solid #aaaaaa;" | style="text-align: center" | 10 | 3,628,800 <span style="font-size:80%; float:right;">≈3.63×10<sup>6</sup></span> | 1,334,961 <span style="font-size:80%; float:right;">≈1.33×10<sup>6</sup></span> |align="right"| β0.36787 94643 |- | style="text-align: center" | 11 | 39,916,800 <span style="font-size:80%; float:right;">≈3.99×10<sup>7</sup></span> | 14,684,570 <span style="font-size:80%; float:right;">≈1.47×10<sup>7</sup></span> |align="right"| β0.36787 94392 |- | style="text-align: center" | 12 | 479,001,600 <span style="font-size:80%; float:right;">≈4.79×10<sup>8</sup></span> | 176,214,841 <span style="font-size:80%; float:right;">≈1.76×10<sup>8</sup></span> |align="right"| β0.36787 94413 |- | style="text-align: center" | 13 | 6,227,020,800 <span style="font-size:80%; float:right;">≈6.23×10<sup>9</sup></span> | 2,290,792,932 <span style="font-size:80%; float:right;">≈2.29×10<sup>9</sup></span> |align="right"| β0.36787 94412 |- | style="text-align: center" | 14 | 87,178,291,200 <span style="font-size:80%; float:right;">≈8.72×10<sup>10</sup></span> | 32,071,101,049 <span style="font-size:80%; float:right;">≈3.21×10<sup>10</sup></span> |align="right"| β0.36787 94412 |-style="border-top:2px solid #aaaaaa;" | style="text-align: center" | 15 |style="font-size:80%;"| 1,307,674,368,000 <span style="float:right;">≈1.31×10<sup>12</sup></span> |style="font-size:80%;"| 481,066,515,734 <span style="float:right;">≈4.81×10<sup>11</sup></span> |align="right"| β0.36787 94412 |- | style="text-align: center" | 16 |style="font-size:80%;"| 20,922,789,888,000 <span style="float:right;">≈2.09×10<sup>13</sup></span> |style="font-size:80%;"| 7,697,064,251,745 <span style="float:right;">≈7.70×10<sup>12</sup></span> |align="right"| β0.36787 94412 |- | style="text-align: center" | 17 |style="font-size:80%;"| 355,687,428,096,000 <span style="float:right;">≈3.56×10<sup>14</sup></span> |style="font-size:80%;"| 130,850,092,279,664 <span style="float:right;">≈1.31×10<sup>14</sup></span> |align="right"| β0.36787 94412 |- | style="text-align: center" | 18 |style="font-size:80%;"| 6,402,373,705,728,000 <span style="float:right;">≈6.40×10<sup>15</sup></span> |style="font-size:80%;"| 2,355,301,661,033,953 <span style="float:right;">≈2.36×10<sup>15</sup></span> |align="right"| β0.36787 94412 |- | style="text-align: center" | 19 |style="font-size:80%;"| 121,645,100,408,832,000 <span style="float:right;">≈1.22×10<sup>17</sup></span> |style="font-size:80%;"| 44,750,731,559,645,106 <span style="float:right;">≈4.48×10<sup>16</sup></span> |align="right"| β0.36787 94412 |-style="border-top:2px solid #aaaaaa;" | style="text-align: center" | 20 |style="font-size:80%;"| 2,432,902,008,176,640,000 <span style="float:right;">≈2.43×10<sup>18</sup></span> |style="font-size:80%;"| 895,014,631,192,902,121 <span style="float:right;">≈8.95×10<sup>17</sup></span> |align="right"| β0.36787 94412 |- | style="text-align: center" | 21 |style="font-size:80%;"| 51,090,942,171,709,440,000 <span style="float:right;">≈5.11×10<sup>19</sup></span> |style="font-size:80%;"| 18,795,307,255,050,944,540 <span style="float:right;">≈1.88×10<sup>19</sup></span> |align="right"| β0.36787 94412 |- | style="text-align: center" | 22 |style="font-size:80%;"| 1,124,000,727,777,607,680,000 <span style="float:right;">≈1.12×10<sup>21</sup></span> |style="font-size:80%;"| 413,496,759,611,120,779,881 <span style="float:right;">≈4.13×10<sup>20</sup></span> |align="right"| β0.36787 94412 |- | style="text-align: center" | 23 |style="font-size:80%;"| 25,852,016,738,884,976,640,000 <span style="float:right;">≈2.59×10<sup>22</sup></span> |style="font-size:80%;"| 9,510,425,471,055,777,937,262 <span style="float:right;">≈9.51×10<sup>21</sup></span> |align="right"| β0.36787 94412 |- | style="text-align: center" | 24 |style="font-size:80%;"| 620,448,401,733,239,439,360,000 <span style="float:right;">≈6.20×10<sup>23</sup></span> |style="font-size:80%;"| 228,250,211,305,338,670,494,289 <span style="float:right;">≈2.28×10<sup>23</sup></span> |align="right"| β0.36787 94412 |-style="border-top:2px solid #aaaaaa;" | style="text-align: center" | 25 |style="font-size:80%;"| 15,511,210,043,330,985,984,000,000 <span style="float:right;">≈1.55×10<sup>25</sup></span> |style="font-size:80%;"| 5,706,255,282,633,466,762,357,224 <span style="float:right;">≈5.71×10<sup>24</sup></span> |align="right"| β0.36787 94412 |- | style="text-align: center" | 26 |style="font-size:80%;"| 403,291,461,126,605,635,584,000,000 <span style="float:right;">≈4.03×10<sup>26</sup></span> |style="font-size:80%;"| 148,362,637,348,470,135,821,287,825 <span style="float:right;">≈1.48×10<sup>26</sup></span> |align="right"| β0.36787 94412 |- | style="text-align: center" | 27 |style="font-size:80%;"| 10,888,869,450,418,352,160,768,000,000 <span style="float:right;">≈1.09×10<sup>28</sup></span> |style="font-size:80%;"| 4,005,791,208,408,693,667,174,771,274 <span style="float:right;">≈4.01×10<sup>27</sup></span> |align="right"| β0.36787 94412 |- | style="text-align: center" | 28 |style="font-size:80%;"| 304,888,344,611,713,860,501,504,000,000 <span style="float:right;">≈3.05×10<sup>29</sup></span> |style="font-size:80%;"| 112,162,153,835,443,422,680,893,595,673 <span style="float:right;">≈1.12×10<sup>29</sup></span> |align="right"| β0.36787 94412 |- | style="text-align: center" | 29 |style="font-size:80%;"| 8,841,761,993,739,701,954,543,616,000,000 <span style="float:right;">≈8.84×10<sup>30</sup></span> |style="font-size:80%;"| 3,252,702,461,227,859,257,745,914,274,516 <span style="float:right;">≈3.25×10<sup>30</sup></span> |align="right"| β0.36787 94412 |-style="border-top:2px solid #aaaaaa;" | style="text-align: center" | 30 |style="font-size:80%;"| 265,252,859,812,191,058,636,308,480,000,000 <span style="float:right;">≈2.65×10<sup>32</sup></span> |style="font-size:80%;"| 97,581,073,836,835,777,732,377,428,235,481 <span style="float:right;">≈9.76×10<sup>31</sup></span> |align="right"| β0.36787 94412 |}
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