# Factorial

The function defined on the set of non-negative integers with value at equal to the product of the natural numbers from 1 to , that is, to ; it is denoted by (by definition, ). For large an approximate expression for the factorial is given by the Stirling formula. The factorial is equal to the number of permutations of elements. The more general expression

is also called a factorial, where is a complex number, is a natural number, and . See also Gamma-function.

#### Comments

Because equals the number of permutations of elements, the factorial is extensively used in combinatorics, probability theory, mathematical statistics, etc. Cf. Combinatorial analysis; Combination; Binomial coefficients.

**How to Cite This Entry:**

Factorial.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Factorial&oldid=18764