Rule of product

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

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

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 sets, 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 are in {A, B, C}, is 3 × 3 = 9.

As another example, when you decide to order pizza, you must first choose the type of crust: thin or deep dish (2 choices). Next, you choose one 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.

In set theory, this multiplication principle is often taken to be the definition of the product of cardinal numbers.[1] We have

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

where ${\displaystyle \times }$ 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.

An extension of the rule of product considers there are n different types of objects, say sweets, to be associated with k objects, say people. How many different ways can the people receive their sweets?

Each person may receive any of the n sweets available, and there are k people, so there are ${\displaystyle \overbrace {n\cdots \cdot n} ^{k}=n^{k}}$ ways to do this.

The rule of sum is another basic counting principle. Stated simply, it is the idea that if we have a ways of doing something and b ways of doing another thing and we can not do both at the same time, then there are a + b ways to choose one of the actions.[3]

• Combinatorial principles