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  "body": "\n## Statement\n\nLet $A$ and $B$ be finite sets with $|A| = m$ and $|B| = n$.\nLet $B^A$ be the set of all functions from $A$ to $B$.\nThen $|B^A| = n^m$.\n\n## Remarks\n\nThe case $m = 0$ gives $n^0 = 1$: there is exactly one function from the empty set to $B$, also when $B$ is empty.\nA function from $A$ to $B$ is a choice of one value in $B$ for each of the $m$ elements of $A$. The [[rule of product|0VPK4X73Q0]] then gives $n^m$.\nFor a subset count, take $B = \\{0, 1\\}$. This gives the [[number of subsets of a finite set|0C5RDJE73X]].\n\n## Explanations\n\n### L1 human\n\nEach element of $A$ independently chooses its image among the $n$ elements of $B$.\nFor example, there are $3^2 = 9$ functions from $\\{1, 2\\}$ to $\\{a, b, c\\}$, because $1$ has 3 choices and $2$ has 3 choices.\n",
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