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The factorial of a non-negataive integer n, designated n!, can be defined recursively as

0! = 1
n! = n(n – 1)! for n > 0.

Intuitively, for a positive integer n, n! is the product of positive integers up to and including n.

Sometimes it's more useful to think of n! as the product of positive integers from n down to 1. That viewpoint, for example, makes clear the relationship

n! = n(n – 1)!

Factorials are used in counting numbers of permutations and combinations.

The gamma function is defined for all real numbers, but outputs factorial values for non-negative integer input. Information about the gamma function can be found in advanced calculus texts.

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