Rule of product

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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.