Rule of product

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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 intuitive 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.[1][2]


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

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.

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 ways to do this.

Related concepts[edit]

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]

See also[edit]


  1. ^ a b Johnston, William, and Alex McAllister. A transition to advanced mathematics. Oxford Univ. Press, 2009. Section 5.1
  2. ^ "College Algebra Tutorial 55: Fundamental Counting Principle". Retrieved December 20, 2014.
  3. ^ Rosen, Kenneth H., ed. Handbook of discrete and combinatorial mathematics. CRC pres, 1999.