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  "body": "\n## Statement\n\nLet $n$ be a natural number, and let $D_n$ be the number of [[derangements|0TH61JR04G]] of an $n$-element set.\nThen\n$$D_n = n! \\sum_{k=0}^{n} \\frac{(-1)^k}{k!} = \\sum_{k=0}^{n} (-1)^k \\frac{n!}{k!}.$$\n\n## Remarks\n\nFor $n = 4$: $24 - 24 + 12 - 4 + 1 = 9$.\nThe proof applies the [[inclusion–exclusion principle|0FF7726C7P]] to the sets $A_i$ of permutations that fix $i$.\nThe sum is a partial sum of the series of $e^{-1}$. Thus $D_n / n! \\to 1/e$, and for $n \\ge 1$, $D_n$ is the integer nearest to $n!/e$.\n\n## Explanations\n\n### L1 human\n\nStart with all $n!$ permutations. Remove those that fix some point, add back those that fix two points, and so on.\nThe result is the alternating sum in the formula.\n",
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