Derangement

Definition

Let $X$ be a set. A derangement of $X$ is a permutation $\sigma$ of $X$ such that $\sigma(x) \neq x$ for all $x \in X$. In other words, it is a permutation with no fixed point.

Remarks

The derangements of $\{1, 2, 3\}$ are $231$ and $312$ in one-line notation. The number of derangements of an $n$-element set is written $D_n$ or $!n$.

Explanations

L1 human

A derangement is a shuffle in which nothing stays in its place. Suppose that $n$ people leave their hats, and each person later takes a hat at random. A derangement is an outcome in which nobody gets their own hat.