Definition:Derangement

Definition
A derangement is a permutation $$f: S \to S$$ from a set $$S$$ to itself where $$f \left({s}\right) \ne s$$ for any $$s \in S$$.

If $$S$$ is finite, the number of derangements is denoted by $$D_n$$ where $$n = \left|{S}\right|$$ (the cardinality of $$S$$.)

Also see

 * Recurrence Relation for the Number of Derangements on a Finite Set

which is shown to be:


 * $$D_n = \begin{cases}

0 & : n = 1 \\ 1 & : n = 2 \\ n \left({D_{n-1} + D_{n-2}}\right) & : n > 2 \end{cases}$$