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### rule of product

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rule of product
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(File:Multiplication-principle.svg|thumb|upright|The elements of the set {A, B} can combine with the elements of the set {1, 2, 3} in six different ways.)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 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

begin{matrix}& underbrace{ left{A,B,Cright} }& & underbrace{ left{ X,Yright} } 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 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.Other typical example is using it with the rule of sum, in this case we have two groups, the group A with 3 elements and the group B with 10 elements. We want to pick one element ( we don't care if it is from group A or B) and a second element that must be from the group B. The way that we can chose the elements are:
mathrm{Total} mathrm{ways} = (3*10) + (10*9)First, we use the rule of product to get the number of ways if we pick a element from group A and then from group B. After this, we repeat the process but now changing the element of group A by element of group B and multiply by the number of element in B - 1 because we pick one of this.

## 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.

## References

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