For any non-negative integer $n$ , the 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 complex values except the negative integers.