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  "body": "\n## Statement\n\nLet $S(n,m)$ be the number of [[surjective functions|01RVCMSN5N]] from the set $[n] = \\{1, \\dots, n\\}$ onto the set $[m] = \\{1, \\dots, m\\}$.\nThen\n$$S(n,m) = \\sum_{k=0}^{m} (-1)^k \\binom{m}{k} (m-k)^n,$$\nwhere $\\binom{m}{k}$ is a [[binomial coefficient|0GBZM0MT0F]].\n\n## Remarks\n\nThe term $(m-k)^n$ is the number of all functions from $[n]$ to a fixed set of $m-k$ elements.\nThe sum comes from the inclusion-exclusion principle: count all functions, then correct for the functions that miss some chosen set of $k$ values of $[m]$.\n",
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  "name": "Formula for the number of surjections between finite sets",
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