# Factorial

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

- (Note: this number is 82 digits long with 14 terminal zeroes!)
- (Note: This number is 2568 digits long and has as much as 249 terminal zeroes!)
- is 38660 digits long and has 2499 terminal zeroes!
- is 456574 digits long and has 24999 terminal zeroes!
- is 973751 digits long and has 49998 terminal zeroes!

## Additional Information

By convention and rules of an empty product, 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

- , where and are positive integers and is as large as possible. Find the value of .

(Source)

- 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)

## See Also

- A cool link to calculate factorials: http://www.nitrxgen.net/factorialcalc.php

On that link, you can calculate factorials from to as much as