Rule of product

Applications

In set theory, this multiplication principle is often taken to be the definition of the product of cardinal numbers. We have

${\displaystyle |S_1|\cdot |S_2|\cdots |S_n|=|S_1\times S_2\times \cdots \times S_n|}$

where ${\displaystyle \times }$ is the Cartesian product operator. These sets need not be finite, nor is it necessary to have only finitely many factors in the product.

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 ${\displaystyle \stackrel{k}{\overbrace{n\cdots \cdot n}}=n^k}$ ways to do this.

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.