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  "body": "\n## Statement\n\nLet $n$ be a natural number, and let $S$ be a subset of $[n]=\\{1,\\dots,n\\}$.\nLet $N(S)$ be the number of [[permutations|0BQMWCNS8S]] of $[n]$ that fix every element of $S$.\nThen $N(S)$ depends only on the size of $S$.\nIf $|S|=k$, then $N(S)=(n-k)!$.\n\n## Remarks\n\nA permutation that fixes each point of $S$ is determined by its values on the other $n-k$ points.\nThese values form an arbitrary permutation of the $n-k$ remaining points.\nThis count is the input for the inclusion-exclusion count of [[derangements|0TH61JR04G]].\n",
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        "locator": "source/s_inclusion-exclusion_derangements.ptx:42",
        "url": "https://github.com/mitchkeller/applied-combinatorics/blob/ebd95d6bd59e23c633c65cf560bcb93efa912aae/source/s_inclusion-exclusion_derangements.ptx#L42"
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  "msc": [
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  "name": "Permutations fixing a given set of points",
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