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

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

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.