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  "body": "\n## Statement\n\nLet $m$ and $n$ be positive integers. Put $[m]=\\{1,\\dots,m\\}$ and $[n]=\\{1,\\dots,n\\}$.\n\nFor a subset $S\\subseteq[m]$, let $N(S)$ be the number of functions from $[n]$ to $[m]$ whose [[image|0Q3F20XB0A]] contains no element of $S$.\n\nThen $N(S)$ depends only on the [[cardinality|05099E1NMJ]] of $S$. If $|S|=k$, then\n$$N(S)=(m-k)^n.$$\n\n## Remarks\n\nA function that avoids $S$ is the same as a function from $[n]$ to the set $[m]\\setminus S$, which has $m-k$ elements. The [[count of functions between finite sets|08QQH9F7G0]] gives $(m-k)^n$.\n\nFor $k=m$ the value is $0$, since $n\\ge 1$ and there is no function into the empty set.\n",
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