Difference between revisions of "Factorial"
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* <math>49! = 608281864034267560872252163321295376887552831379210240000000000</math> | * <math>49! = 608281864034267560872252163321295376887552831379210240000000000</math> | ||
* <math>50! = 30414093201713378043612608166064768844377641568960512000000000000</math> | * <math>50! = 30414093201713378043612608166064768844377641568960512000000000000</math> | ||
− | + | * <math>51! = 1551118753287382280224243016469303211063259720016986112000000000000</math> | |
+ | * <math>52! = 80658175170943878571660636856403766975289505440883277824000000000000</math> | ||
+ | * <math>53! = 4274883284060025564298013753389399649690343788366813724672000000000000</math> | ||
+ | * <math>54! = 230843697339241380472092742683027581083278564571807941132288000000000000</math> | ||
+ | * <math>55! = 12696403353658275925965100847566516959580321051449436762275840000000000000</math> | ||
+ | * <math>56! = 710998587804863451854045647463724949736497978881168458687447040000000000000</math> | ||
+ | * <math>57! = 40526919504877216755680601905432322134980384796226602145184481280000000000000</math> | ||
+ | * <math>58! = 2350561331282878571829474910515074683828862318181142924420699914240000000000000</math> | ||
+ | * <math>59! = 138683118545689835737939019720389406345902876772687432540821294940160000000000000</math> | ||
+ | * <math>60! = 8320987112741390144276341183223364380754172606361245952449277696409600000000000000</math> (Note: this number is 82 digits long!) | ||
== Additional Information == | == Additional Information == |
Revision as of 23:20, 29 March 2011
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 .
Examples
- (Note: this number is 82 digits long!)
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)
See Also
- A cool link to calculate factorials: http://www.nitrxgen.net/factorialcalc.php