In mathematics, the factorial of a positive integer n, denoted n!, is the product of the positive integers less than or equal to n. For example,
Usually, n! is read as "n factorial". The current notation was introduced by the mathematician Christian Kramp[?] in 1808.
Factorials are often used as a simple example when teaching recursion in computer science because they satisfy the following recursive relationship
Factorials are important in combinatorics because there are n! different ways of arranging n distinct objects in a sequence (see permutation). They also turn up in formulas of calculus, such as in Taylor's theorem, for instance, because the nth derivative of the function x^{n} is n!.
When n is large, n! can be estimated quite accurately using Stirling's approximation

The related Gamma function Γ(z) is defined for all complex numbers z except for z = 0, 1, 2, 3, ... It is related to the factorial by the property:
when n is any nonnegative integer.
A common related notation is to use multiple exclamation points (!) to denote a multifactorial, the product of integers in steps of two, three, or more.
For example, n!! denotes the double factorial of n, defined recursively by n!! = n (n2)!! for n > 1 and as 1 for n = 0,1. Thus, (2n)!! = 2^{n}n! and (2n+1)! = (2n+1)!! 2^{n}n!. The double factorial is related to the Gamma function of halfinteger order by Γ(n+1/2) = √π (2n1)!!/2^{n}.
One should be careful not to interpret n!! as the factorial of n!, a much larger number.
The double factorial is the most commonly used variant, but one can similarly define the triple factorial (!!!) and so on. In general, the kth factorial, denoted by !^{(k)}, is defined recursively by: n!^{(k)} = n (nk)!^{(k)} for n > k1, n!^{(k)} = n for k > n > 0, and 0!^{(k)} = 1.
Occasionally the hyperfactorial of n is considered. It is written as H(n) and defined by
E.g. H(4) = 27648.
The hyperfactorial function is similar to the factorial, but produces larger numbers. The rate of growth of this function, however, is not much larger than a regular factorial.
The superfactorial of n, written as n$ (a factorial sign with an S written over it) has been defined as
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