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  "body": "\n## Statement\n\nLet $X$ be a finite set with $|X| = n$.\nThen the number of [[derangements|0TH61JR04G]] of $X$ is the derangement number $D_n$ of the node [[number of derangements|03WY4HHMT1]].\n\n## Remarks\n\nThe proof checks the recursion $D_{n+2} = (n + 1)(D_n + D_{n+1})$, as in the node [[recurrence for derangements|0QRRYXG7V7]].\nFor $n = 4$ there are $9$ derangements among the $24$ permutations.\n\n## Explanations\n\n### L1 human\n\nThe derangement numbers are defined by a recursion, and this result says that they really count the permutations with no fixed point.\nFor example, $D_3 = 2$, and $\\{1, 2, 3\\}$ has exactly the two derangements $231$ and $312$.\n",
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  "name": "The derangement numbers count derangements",
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