In mathematics, what is a factorial?

A factorial is defined as the product of an integer and all the positive integers less than that integer. For example, the factorial of 5, denoted as 5!, is calculated as 5 × 4 × 3 × 2 × 1, which equals 120. This operation is essential in various fields of mathematics, particularly in combinatorics, where it is used to calculate permutations and combinations.

This definition captures the essence of what a factorial represents—a cumulative product that emphasizes the role of multiplication in building the factorial value. Understanding factorials is crucial for solving problems related to arrangements and selections in mathematical contexts, making it a significant concept in quantitative analysis and business applications. The other options do not define factorials accurately; they refer to different mathematical operations such as sums, differences, and quotients.