# Difference between revisions of "Factorial"

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* <math>28! = 304888344611713860501504000000</math> | * <math>28! = 304888344611713860501504000000</math> | ||

* <math>29! = 8841761993739701954543616000000</math> | * <math>29! = 8841761993739701954543616000000</math> | ||

+ | * <math>30! = 265252859812191058636308480000000</math> | ||

== Additional Information == | == Additional Information == |

## Revision as of 19:48, 29 March 2011

The **factorial** is an important function in combinatorics and analysis, used to determine the number of ways to arrange objects.

## Contents

## Definition

The factorial is defined for positive integers as . Alternatively, a recursive definition for the factorial is .

## Examples

## 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

*Main article: 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 converge to rapidly, as it is the reciprocal of an exponential function. 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.

## Problems

### Introductory

- Find the units digit of the sum

(Source)

### Intermediate

- Let be the product of the first positive odd integers. Find the largest integer such that is divisible by

(Source)

### Olympiad

- Let be the number of permutations of the set , which have exactly fixed points. Prove that
.

(Source)