# Factorial

## Factorial

For any positive integer n, a number that is the product of all the positive integers less than or equal to n.

• In set theory, the factorial n is defined as being the number of permutations of a set of n elements.
• We say that “n!” is a function of the set of whole numbers in itself, defined like this: ∀ $$n\in \mathbb{N}\space\vert\ n! = 1 \times 2 \times 3 \times \ldots \times n$$.
• By convention, 0! = 1, because there is only one permutation of a set that contains 0 elements.

### Symbol

• The expression “5 factorial” is noted as “5!”.
• The symbol is “!” which can be associated with the exclamation point.

### Examples

• 5! = 5 × 4 × 3 × 2 × 1 =  120
• 4! = 4 × 3 × 2 × 1 =  24

### Historical Note

The French mathematician Christian Kramp (1760-1826) introduced factorial notation in his book Éléments d’Arithmétique Universelle, published in 1808.