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.