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.