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  "body": "\n## Definition\n\nLet $X$ be a set.\nA derangement of $X$ is a [[permutation|0BQMWCNS8S]] $\\sigma$ of $X$ such that $\\sigma(x) \\neq x$ for all $x \\in X$.\nIn other words, it is a permutation with no fixed point.\n\n## Remarks\n\nThe derangements of $\\{1, 2, 3\\}$ are $231$ and $312$ in one-line notation.\nThe [[number of derangements|03WY4HHMT1]] of an $n$-element set is written $D_n$ or $!n$.\n\n## Explanations\n\n### L1 human\n\nA derangement is a shuffle in which nothing stays in its place.\nSuppose 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.\n",
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