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

$\begin{array}{cccc}& \underbrace{\{A,B,C\}}& & \underbrace{\{X,Y\}}\\ \mathrm{T}\mathrm{o}\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{o}\mathrm{f}& \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{s}\mathrm{e}& \mathrm{A}\mathrm{N}\mathrm{D}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{o}\mathrm{f}& \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{s}\mathrm{e}\end{array}$

$\begin{array}{cc}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{o}\mathrm{f}& \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{s}\mathrm{e}.\\ & \overbrace{\{AX,AY,BX,BY,CX,CY\}}\end{array}$

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

$\left|S_1\right|\cdot \left|S_2\right|\cdots \left|S_n\right|=|S_1\times S_2\times \cdots \times S_n|$

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.

example

When you decide to order pizza, you must first choose the type of crust: thin or deep dish (2 choices). Next, you choose the topping: cheese, pepperoni, or sausage (3 choices).

Using the rule of product, you know that there are 2 × 3 = 6 possible combinations of ordering a pizza.

See also

• Combinatorial principle
• Rule of sum

References

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