factorial

For any non-negative integer $n$, the factorial of $n$, denoted $n!$, can be defined by

$$n!=\prod _{r=1}^{n}r$$

where for $n=0$ the empty product is taken to be $1$.

Alternatively, the factorial can be defined recursively by $0!=1$ and $n!=n(n-1)!$ for $n>0$.

$n!$ is equal to the number of permutations of $n$ distinct objects. For example, there are $5!$ ways to arrange the five letters A, B, C, D and E into a word.

For every non-negative integer $n$ we have

$$\mathrm{\Gamma }(n+1)=n!$$

where $\mathrm{\Gamma }$ is Euler’s gamma function. In this way the notion of factorial can be generalized to all complex (http://planetmath.org/Complex) values except the negative integers.

Title	factorial
Canonical name	Factorial
Date of creation	2013-03-22 11:53:58
Last modified on	2013-03-22 11:53:58
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	22
Author	yark (2760)
Entry type	Definition
Classification	msc 05A10
Classification	msc 11B65
Classification	msc 92-01
Classification	msc 92B05
Synonym	factorial function
Related topic	BinomialCoefficient
Related topic	ExponentialFactorial