Probability of a permutation being a derangement is 1/e (in the limit as n-->Infinity)    

n    n! Permutations   !n  Derangements     !n/n! --> 1/e

1                  1                 0        0.000000000
2                  2                 1        0.500000000
3                  6                 2        0.333333333
4                 24                 9        0.375000000
5                120                44        0.366666667
6                720               265        0.368055556
7              5,040             1,854        0.367857143
8             40,320            14,833        0.367881944
9            362,880           133,496        0.367879189
10         3,628,800         1,334,961        0.367879464
11        39,916,800        14,684,570        0.367879439
12       479,001,600       176,214,841        0.367879441
13     6,227,020,800     2,290,792,932        0.367879441
14    87,178,291,200    32,071,101,049        0.367879441
15 1,307,674,368,000   481,066,515,734        0.367879441