# Factorial

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,

- 5! = 5 × 4 × 3 × 2 × 1 = 120
- 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800

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 (if

*n* ≥ 1):

*n*! = *n* (*n* − 1)!

In addition, one defines

- 0! = 1

for several related reasons:

- 0! is an instance of the empty product, and therefore 1
- it makes the above recursive relation work for
*n* = 1
- many identities in combinatorics would not work for zero sizes without this definition

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

*n*-th derivative of the function

*x*^{n} is

*n*!.

When *n* is large, *n*! can be estimated quite accurately using Stirling's approximation:

A simple online factorial calculator can be obtained

here.

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 non-negative 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* (*n*-2)!! for *n* > 1 and as 1 for *n* = 0,1. Thus, (2*n*)!! = 2^{n}*n*! and (2*n*+1)! = (2*n*+1)!! 2^{n}*n*!. The double factorial is related to the Gamma function of half-integer order by Γ(*n*+1/2) = √π (2*n*-1)!!/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 *k*-th factorial, denoted by !^{(k)}, is defined recursively by: n!^{(k)} = n (n-k)!^{(k)} for *n* > *k*-1, 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

*H*(*n*) = *n*^{n} (*n*-1)^{(n-1)} ... 3^{3} 2^{2} 1^{1}

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
*n*$ = *n*!^{(4)}*n*!

where the ^{(4)} notation denotes the hyper4 operator, or using

Knuth's up-arrow notation,

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