For alternative uses of the exclamation point symbol, please see double factorial for expressionsof the form4!!or subfactorial for expressions of the form!3.

**Factorial** is a function denoted by a trailing exclamation point (!), which is defined for all non-negative integers.

For any positive integer, it outputs the product of all natural numbers between 1 and that number, inclusive:

The notation is read " factorial". Alternatively, one could think of the product as being in the opposite order:

As a consequence of the empty product,

As a concrete example:

Factorials are commonly used in combinatorics and probability theory. It is also used in Taylor polynomials and infinite series.

The factorial function can also be seen as a specific case of the gamma function (), which extends the factorial to the complex plane (excluding the non-positive integers). In particular, for all values for which the factorial is defined: