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A derangement is a permutation $f: S \to S$ from a set $S$ to itself where $\map f s \ne s$ for any $s \in S$.

If $S$ is finite, the number of derangements is denoted by $D_n$ where $n = \card S$ (the cardinality of $S$.)

Also see

Historical Note

The number of a derangements of a finite set was first investigated by Nicolaus I Bernoulli and Pierre Raymond de Montmort between about $1708$ and $1713$.

They solved it at around the same time.

The question is often couched in the idea of counting the number of ways of placing letters at random in envelopes such that nobody receives the correct letter.