Number of functions between finite sets

Statement

Let $A$ and $B$ be finite sets with $|A| = m$ and $|B| = n$. Let $B^A$ be the set of all functions from $A$ to $B$. Then $|B^A| = n^m$.

Remarks

The case $m = 0$ gives $n^0 = 1$: there is exactly one function from the empty set to $B$, also when $B$ is empty. A 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 then gives $n^m$. For a subset count, take $B = \{0, 1\}$. This gives the number of subsets of a finite set.

Explanations

L1 human

Each element of $A$ independently chooses its image among the $n$ elements of $B$. For example, there are $3^2 = 9$ functions from $\{1, 2\}$ to $\{a, b, c\}$, because $1$ has 3 choices and $2$ has 3 choices.