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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. 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.

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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. 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 where 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.