structural:reformatting
Applications

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

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 n ⋯ ⋅ n ⏞ k = n k {\displaystyle \overbrace {n\cdots \cdot n} ^{k}=n^{k}} ways to do this.