Difference between revisions of "Factorial"
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=== Examples === | === Examples === | ||
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* [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=508851#p508851 AIME 2003I/1] | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=508851#p508851 AIME 2003I/1] |
Revision as of 13:35, 24 July 2006
The factorial is an important function in combinatorics and analysis, used to determine the number of ways to arrange objects.
Contents
[hide]Definition
The factorial is defined for positive integers as . Alternatively, a recursive definition for the factorial is: .
Additional Information
By convention, is given the value .
The gamma function is a generalization of the factorial to values other than nonnegative integers.
Prime factorization
Since is the product of all positive integers not exceeding , it is clear that it is divisible by all primes , and not divisible by any prime . But what is the power of a prime in the prime factorization of ? We can find it as the sum of powers of in all the factors ; but rather than counting the power of in each factor, we shall count the number of factors divisible by a given power of . Among the numbers , exactly are divisible by (here is the floor function). The ones divisible by give one power of . The ones divisible by give another power of . Those divisible by give yet another power of . Continuing in this manner gives
for the power of in the prime factorization of . The series is formally infinite, but the terms become pretty fast. For example, the power of in is just ( is already greater than ).
Uses
The factorial is used in the definitions of combinations and permutations, as is the number of ways to order distinct objects.
Examples
- 2006 AIME II Problem 3 on finding prime powers in a factorial