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−  The function defined on the set of nonnegative integers with value at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038080/f0380801.png" /> equal to the product of the natural numbers from 1 to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038080/f0380802.png" />, that is, to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038080/f0380803.png" />; it is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038080/f0380804.png" /> (by definition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038080/f0380805.png" />). For large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038080/f0380806.png" /> an approximate expression for the factorial is given by the [[Stirling formulaStirling formula]]. The factorial is equal to the number of permutations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038080/f0380807.png" /> elements. The more general expression  +  {{TEXdone}} 
 +  The function defined on the set of nonnegative integers with value at $n$ equal to the product of the natural numbers from 1 to $n$, that is, to $1\cdot2\ldots n$; it is denoted by $n!$ (by definition, $0!=1$). For large $n$ an approximate expression for the factorial is given by the [[Stirling formulaStirling formula]]. The factorial is equal to the number of permutations of $n$ elements. The more general expression 
   
−  <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;textalign:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038080/f0380808.png" /></td> </tr></table>
 +  $$(a)_\mu=a(a+1)\ldots(a+\mu1),$$ 
   
−  is also called a factorial, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038080/f0380809.png" /> is a complex number, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038080/f03808010.png" /> is a natural number, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038080/f03808011.png" />. See also [[GammafunctionGammafunction]].  +  is also called a factorial, where $a$ is a complex number, $\mu$ is a natural number, and $(a)_0=1$. See also [[GammafunctionGammafunction]]. 
   
   
   
 ====Comments====   ====Comments==== 
−  Because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038080/f03808012.png" /> equals the number of permutations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038080/f03808013.png" /> elements, the factorial is extensively used in combinatorics, probability theory, mathematical statistics, etc. Cf. [[Combinatorial analysisCombinatorial analysis]]; [[CombinationCombination]]; [[Binomial coefficientsBinomial coefficients]].  +  Because $n!$ equals the number of permutations of $n$ elements, the factorial is extensively used in combinatorics, probability theory, mathematical statistics, etc. Cf. [[Combinatorial analysisCombinatorial analysis]]; [[CombinationCombination]]; [[Binomial coefficientsBinomial coefficients]]. 
Revision as of 15:28, 9 April 2014
The function defined on the set of nonnegative integers with value at $n$ equal to the product of the natural numbers from 1 to $n$, that is, to $1\cdot2\ldots n$; it is denoted by $n!$ (by definition, $0!=1$). For large $n$ an approximate expression for the factorial is given by the Stirling formula. The factorial is equal to the number of permutations of $n$ elements. The more general expression
$$(a)_\mu=a(a+1)\ldots(a+\mu1),$$
is also called a factorial, where $a$ is a complex number, $\mu$ is a natural number, and $(a)_0=1$. See also Gammafunction.
Because $n!$ equals the number of permutations of $n$ elements, the factorial is extensively used in combinatorics, probability theory, mathematical statistics, etc. Cf. Combinatorial analysis; Combination; Binomial coefficients.
How to Cite This Entry:
Factorial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Factorial&oldid=18764
This article was adapted from an original article by BSE3 (originator), which appeared in Encyclopedia of Mathematics  ISBN 1402006098.
See original article