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


 * Closed Form of Number of Derangements on a Finite Set