In combinatorics, the **rule of product** is a basic counting principle (a.k.a. the fundamental principle of counting). Stated simply, it is the idea that if we have *a* ways of doing something and *b* ways of doing another thing, then there are *a* · *b* ways of performing both actions.

- $ \begin{matrix} & \underbrace{ \left\{A,B,C\right\} } & & \underbrace{ \left\{ X,Y\right\} } \\ \mathrm{To}\ \mathrm{choose}\ \mathrm{one}\ \mathrm{of} & \mathrm{these} & \mathrm{AND}\ \mathrm{one}\ \mathrm{of} & \mathrm{these} \end{matrix} $

- $ \begin{matrix} \mathrm{is}\ \mathrm{to}\ \mathrm{choose}\ \mathrm{one}\ \mathrm{of} & \mathrm{these}. \\ & \overbrace{ \left\{ AX, AY, BX, BY, CX, CY \right\} } \end{matrix} $

In this example, the rule says: multiply 3 by 2, getting 6.

The sets {*A*, *B*, *C*} and {*X*, *Y*} in this example are disjoint, but that is not necessary. The number of ways to choose a member of {*A*, *B*, *C*}, and then to do so again, in effect choosing an ordered pair each of whose components is in {*A*, *B*, *C*}, is 3 × 3 = 9.

In set theory, this multiplication principle is often taken to be the definition of the product of cardinal numbers. We have

- $ |S_{1}|\cdot|S_{2}|\cdots|S_{n}| = |S_{1} \times S_{2} \times \cdots \times S_{n}| $

where $ \times $ is the Cartesian product operator. These sets need not be finite, nor is it necessary to have only finitely many factors in the product; see cardinal number.