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*rule of product*

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rule of product

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**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**Â·

**b**ways of performing both actions.Johnston, William, and Alex McAllister.

*A transition to advanced mathematics*. Oxford Univ. Press, 2009. Section 5.1WEB,weblink College Algebra Tutorial 55: Fundamental Counting Principle, December 20, 2014,

## Examples

*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 is 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.

## Applications

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.## Related concepts

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.Rosen, Kenneth H., ed.

*Handbook of discrete and combinatorial mathematics.*CRC pres, 1999.

## See also

## References

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