derangement

In mathematics, a derangement is a permutation of the elements of a set, such that no element appears in its original position. For example, if you have three items labeled 1, 2, and 3, a derangement would be any arrangement where 1 is not in the first position, 2 is not in the second, and 3 is not in the third.

The number of derangements of n items, denoted as D_n, can be approximated by n!/e, where n! is the factorial of n and e is the base of the natural logarithm. This concept is often illustrated with the "hat-check problem": if n people check their hats and the hats are returned randomly, the probability that no one receives their own hat is approximately D_n/n!, which is about 1/e.12