Rule of product

Statement

Let $A$ and $B$ be finite sets. Then the Cartesian product $A \times B = \{(a, b) : a \in A, b \in B\}$ is finite and $|A \times B| = |A| \cdot |B|$.

More generally, for finite sets $A_1, \dots, A_k$, the product $A_1 \times \dots \times A_k$ has $|A_1| \cdots |A_k|$ elements.

Remarks

A common form of the rule: a choice is made in $k$ steps, and step $i$ has $n_i$ options, whatever the earlier steps are. Then there are $n_1 n_2 \cdots n_k$ choices in total. The number of functions between finite sets is a direct consequence.

Explanations

L1 human

If you choose one item from a first group and then one item from a second group, you multiply the sizes. For example, with 3 shirts and 4 trousers there are $3 \cdot 4 = 12$ outfits.