Factorial

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

Factorials Video

Factorials

Definition

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

Examples

• (remember! this is 1, not 0! (the '!' was an exclamation mark, not a factorial sign))
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• (Note: this number is 82 digits long with 14 terminal zeroes!)
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• (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

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Intermediate

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

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

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Olympiad

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

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See Also

• Combinatorics

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

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