Understanding Subfactorials: Definition, Examples, and Applications

The factorial of a number n, denoted by n!, is the product of all positive integers less than or equal to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.

A subfactorial, denoted by n!!, is a special type of factorial that is defined as the product of all positive integers less than or equal to n, except for n itself. In other words, n!! = n × (n-1) × (n-2) × ... × 2 × 1.

For example, 5!! = 5 × 4 × 3 × 2 × 1 = 120, which is the same as 5! because 5 is not included in the product. Similarly, 4!! = 4 × 3 × 2 × 1 = 12, and 3!! = 3 × 2 × 1 = 6.

Subfactorials are used in various areas of mathematics, such as combinatorics, algebra, and number theory. They have applications in counting the number of ways to arrange objects in a particular order, or in finding the number of solutions to a mathematical equation.