factorial Factorial 2001-10-26 03:07:34 2006-10-09 16:34:38 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \PMlinkescapeword{word} For any non-negative integer $n$, the {\em factorial} 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 \PMlinkname{complex}{Complex} values except the negative integers.