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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

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 xn is n!.

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

<math>n!\sim\sqrt{2\pi n}\left(\frac{n}{e}\right)^n.</math>

Table of contents

Generalization to the Gamma function

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:

<math>\Gamma(n+1) = n!</math>

when n is any non-negative integer.

Multifactorials

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, (2n)!! = 2nn! and (2n+1)! = (2n+1)!! 2nn!. The double factorial is related to the Gamma function of half-integer order by Γ(n+1/2) = √π (2n-1)!!/2n.

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.

Hyperfactorials

Occasionally the hyperfactorial of n is considered. It is written as H(n) and defined by

H(n) = nn (n-1)(n-1) ... 33 22 11

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.

Superfactorials

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.

References



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