Rule of product
In combinatorics, the rule of product or multiplication principle is a basic counting principle. Stated simply, it is the intuitive idea that if there are ways of doing something and ways of doing another thing, then there are ways of performing both actions.
Examples
In this example, the rule says: multiply 3 by 2, getting 6.The sets and in this example are disjoint sets, but that is not necessary. The number of ways to choose a member of, and then to do so again, in effect choosing an ordered pair each of whose components are in, 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. Next, you choose one topping: cheese, pepperoni, or sausage.
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 havewhere is the Cartesian product operator. These sets need not be finite, nor is it necessary to have only finitely many factors in the product.
An extension of the rule of product considers there are different types of objects, say sweets, to be associated with objects, say people. How many different ways can the people receive their sweets?
Each person may receive any of the sweets available, and there are people, so there are ways to do this.