Factorial

In mathematics, the factorial of a positive integer n, denoted by n!, is the product of all positive integers less than or equal to n:

${\displaystyle n!=n\times (n-1)\times (n-2)\times (n-3)\times \cdots \times 3\times 2\times 1\,.}$

For example,

${\displaystyle 5!=5\times 4\times 3\times 2\times 1=120\,.}$

The value of 0! is 1, according to the convention for an empty product.

The factorial operation is encountered in many areas of mathematics, notably in combinatorics, algebra, and mathematical analysis. Its most basic use counts the possible distinct sequences – the permutations – of n distinct objects: there are n!.

The factorial function can also be extended to non-integer arguments while retaining its most important properties by defining x! = Γ(x + 1), where Γ is the gamma function; this is undefined when x is a negative integer.