For alternative uses of the exclamation point symbol, please see double factorial for expressions
of the form 4!! 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:

$ n!\equiv1\cdot2\cdot3\cdots n $

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

$ n!\equiv n(n-1)(n-2)\cdots3\cdot2\cdot1 $

As a consequence of the empty product,

$ 0!\equiv1 $

As a concrete example:

$ 5!=1\cdot2\cdot3\cdot4\cdot5=120 $

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 ($ \Gamma $), which extends the factorial to the complex plane (excluding the non-positive integers). In particular, for all values for which the factorial is defined:

$ n!=\Gamma(n+1) $

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