# factorial

For any non-negative integer $n$, the of $n$, denoted $n!$, can be defined by

 $n!=\prod_{r=1}^{n}r$

where for $n=0$ the empty product is taken to be $1$.

Alternatively, the factorial can be defined recursively by $0!=1$ and $n!=n(n-1)!$ for $n>0$.

$n!$ is equal to the number of permutations of $n$ distinct objects. For example, there are $5!$ ways to arrange the five letters A, B, C, D and E into a word.

For every non-negative integer $n$ we have

 $\Gamma(n+1)=n!$

where $\Gamma$ is Euler’s gamma function. In this way the notion of factorial can be generalized to all complex (http://planetmath.org/Complex) values except the negative integers.

 Title factorial Canonical name Factorial Date of creation 2013-03-22 11:53:58 Last modified on 2013-03-22 11:53:58 Owner yark (2760) Last modified by yark (2760) Numerical id 22 Author yark (2760) Entry type Definition Classification msc 05A10 Classification msc 11B65 Classification msc 92-01 Classification msc 92B05 Synonym factorial function Related topic BinomialCoefficient Related topic ExponentialFactorial