# Factorial

The function defined on the set of non-negative integers with value at $n$ equal to the product of the natural numbers from 1 to $n$, that is, to $1\cdot2\cdot\ldots\cdot n$; it is denoted by $n!$ (by definition, $0!=1$). For large $n$ an approximate expression for the factorial is given by the Stirling formula. The factorial is equal to the number of permutations of $n$ elements. The more general expression

$$(a)_\mu=a(a+1)\dots(a+\mu-1) \ ,$$

is also called a factorial, where $a$ is a complex number, $\mu$ is a natural number, and $(a)_0=1$. See also Gamma-function.

#### Comments

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

The *Pochhammer symbol* $(a)_\mu$ denotes the *rising factorial*, also denoted $(a)^{\overline{\mu}}$. Analogously, one defines the *falling factorial*
$$
(a)^{\underline{\mu}} = a(a-1)\dots(a-\mu+1) \ .
$$

**How to Cite This Entry:**

Factorial.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Factorial&oldid=43580