PlanetMath: factorial

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factorial

(Definition)

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

$\displaystyle n!=\prod_{r=1}^n r$

where for

the empty product is taken to be

Alternatively, the factorial can be defined recursively by and for .

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

For every non-negative integer we have

$\displaystyle \Gamma(n+1) = n!$

where $\Gamma$ is Euler's gamma function. In this way the notion of factorial can be generalized to all complex values except the negative integers.

"factorial" is owned by yark. [ full author list (2) | owner history (1) ]

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Other names:

factorial function

Attachments:: the prime power dividing a factorial (Theorem) by Thomas Heye
factorial prime (Definition) by PrimeFan
recursive algorithm for factorial function (Algorithm) by PrimeFan
factorion (Definition) by Mravinci

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Cross-references: negative, Euler's gamma function, objects, permutations, number, empty product, integer
There are 36 references to this entry.

This is version 17 of factorial, born on 2001-10-26, modified 2006-10-09.
Object id is 516, canonical name is Factorial.
Accessed 17324 times total.

Classification:

AMS MSC:	05A10 (Combinatorics :: Enumerative combinatorics :: Factorials, binomial coefficients, combinatorial functions)
	11B65 (Number theory :: Sequences and sets :: Binomial coefficients; factorials; $q$-identities)

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