# Difference between revisions of "Factorial"

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* <math>6! = 720</math> | * <math>6! = 720</math> | ||

* <math>7! = 5040</math> | * <math>7! = 5040</math> | ||

+ | * <math>8! = 40320</math> | ||

+ | * <math>9! = 362880</math> | ||

+ | * <math>10! = 3628800</math> | ||

+ | * <math>11! = 39916800</math> | ||

+ | * <math>12! = 479001600</math> | ||

+ | * <math>13! = 6227020800</math> | ||

+ | * <math>14! = 87178291200</math> | ||

+ | * <math>15! = 1307674368000</math> | ||

+ | * <math>16! = 20922789888000</math> | ||

+ | * <math>17! = 355687428096000</math> | ||

+ | * <math>18! = 6402373705728000</math> | ||

+ | * <math>19! = 121645100408832000</math> | ||

+ | * <math>20! = 2432902008176640000</math> | ||

+ | * <math>21! = 51090942171709440000</math> | ||

+ | * <math>22! = 1124000727777607680000</math> | ||

+ | * <math>23! = 25852016738884976640000</math> | ||

+ | * <math>24! = 620448401733239439360000</math> | ||

+ | * <math>25! = 15511210043330985984000000</math> | ||

+ | * <math>26! = 403291461126605635584000000</math> | ||

+ | * <math>27! = 10888869450418352160768000000</math> | ||

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

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

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

* <math>31! = 8222838654177922817725562880000000</math> | * <math>31! = 8222838654177922817725562880000000</math> | ||

* <math>32! = 263130836933693530167218012160000000</math> | * <math>32! = 263130836933693530167218012160000000</math> |

## Revision as of 21:25, 3 February 2019

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!)
- 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