Difference between revisions of "Factorial"
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The '''factorial''' is an important function in [[combinatorics]] and [[analysis]], used to determine the number of ways to arrange objects. | The '''factorial''' is an important function in [[combinatorics]] and [[analysis]], used to determine the number of ways to arrange objects. | ||
+ | |||
+ | == Factorials Video == | ||
+ | [https://youtu.be/axFmwEI9ddk Factorials] | ||
== Definition == | == Definition == | ||
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== Examples == | == Examples == | ||
− | * <math>0! = 1</math> | + | * <math>0! = 1</math> (remember! this is 1, not 0! (the '!' was an exclamation mark, not a factorial sign)) |
* <math>1! = 1</math> | * <math>1! = 1</math> | ||
* <math>2! = 2</math> | * <math>2! = 2</math> | ||
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* <math>59! = 138683118545689835737939019720389406345902876772687432540821294940160000000000000</math> | * <math>59! = 138683118545689835737939019720389406345902876772687432540821294940160000000000000</math> | ||
* <math>60! = 8320987112741390144276341183223364380754172606361245952449277696409600000000000000</math> (Note: this number is 82 digits long with 14 terminal zeroes!) | * <math>60! = 8320987112741390144276341183223364380754172606361245952449277696409600000000000000</math> (Note: this number is 82 digits long with 14 terminal zeroes!) | ||
− | * <math>1000!</math> is 2568 digits long and has as much as 249 terminal zeroes! | + | * <math>100! = 93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000</math> |
+ | * <math>1000! = 402387260077093773543702433923003985719374864210714632543799910429938512398629020592044208486969404800479988610197196058631666872994808558901323829669944590997424504087073759918823627727188732519779505950995276120874975462497043601418278094646496291056393887437886487337119181045825783647849977012476632889835955735432513185323958463075557409114262417474349347553428646576611667797396668820291207379143853719588249808126867838374559731746136085379534524221586593201928090878297308431392844403281231558611036976801357304216168747609675871348312025478589320767169132448426236131412508780208000261683151027341827977704784635868170164365024153691398281264810213092761244896359928705114964975419909342221566832572080821333186116811553615836546984046708975602900950537616475847728421889679646244945160765353408198901385442487984959953319101723355556602139450399736280750137837615307127761926849034352625200015888535147331611702103968175921510907788019393178114194545257223865541461062892187960223838971476088506276862967146674697562911234082439208160153780889893964518263243671616762179168909779911903754031274622289988005195444414282012187361745992642956581746628302955570299024324153181617210465832036786906117260158783520751516284225540265170483304226143974286933061690897968482590125458327168226458066526769958652682272807075781391858178889652208164348344825993266043367660176999612831860788386150279465955131156552036093988180612138558600301435694527224206344631797460594682573103790084024432438465657245014402821885252470935190620929023136493273497565513958720559654228749774011413346962715422845862377387538230483865688976461927383814900140767310446640259899490222221765904339901886018566526485061799702356193897017860040811889729918311021171229845901641921068884387121855646124960798722908519296819372388642614839657382291123125024186649353143970137428531926649875337218940694281434118520158014123344828015051399694290153483077644569099073152433278288269864602789864321139083506217095002597389863554277196742822248757586765752344220207573630569498825087968928162753848863396909959826280956121450994871701244516461260379029309120889086942028510640182154399457156805941872748998094254742173582401063677404595741785160829230135358081840096996372524230560855903700624271243416909004153690105933983835777939410970027753472000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000</math> (Note: This number is 2568 digits long and has as much as 249 terminal zeroes!) | ||
+ | * <math>10000!</math> is 38660 digits long and has 2499 terminal zeroes! | ||
+ | * <math>100000!</math> is 456574 digits long and has 24999 terminal zeroes! | ||
+ | * <math>200000!</math> is 973751 digits long and has 49998 terminal zeroes! | ||
== Additional Information == | == Additional Information == | ||
− | By [[mathematical convention|convention]], <math>0!</math> is given the value <math>1</math>. | + | By [[mathematical convention|convention]] and rules of an empty product, <math>0!</math> is given the value <math>1</math>. |
The [[gamma function]] is a generalization of the factorial to values other than [[nonnegative integer]]s. | The [[gamma function]] is a generalization of the factorial to values other than [[nonnegative integer]]s. | ||
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==Problems== | ==Problems== | ||
===Introductory=== | ===Introductory=== | ||
− | *Find the | + | *Find the unitsdigit of the sum |
<cmath>\sum_{i=1}^{100}(i!)^{2}</cmath> | <cmath>\sum_{i=1}^{100}(i!)^{2}</cmath> | ||
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<math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,3\quad\mathrm{(D)}\,5\quad\mathrm{(E)}\,7\quad\mathrm{(F)}\,9</math> | <math>\mathrm{(A)}\,0\quad\mathrm{(B)}\,1\quad\mathrm{(C)}\,3\quad\mathrm{(D)}\,5\quad\mathrm{(E)}\,7\quad\mathrm{(F)}\,9</math> | ||
([[2007 iTest Problems/Problem 6|Source]]) | ([[2007 iTest Problems/Problem 6|Source]]) | ||
+ | |||
===Intermediate=== | ===Intermediate=== | ||
+ | *<math>\frac{((3!)!)!}{3!}=k*n!</math>, where <math>k</math> and <math>n</math> are positive integers and <math>n</math> is as large as possible. Find the value of <math>k+n</math>. | ||
+ | ([[2003 AIME I Problems/Problem 1|Source]]) | ||
*Let <math>P </math> be the product of the first <math>100</math> [[positive integer | positive]] [[odd integer]]s. Find the largest integer <math>k </math> such that <math>P </math> is divisible by <math>3^k .</math> | *Let <math>P </math> be the product of the first <math>100</math> [[positive integer | positive]] [[odd integer]]s. Find the largest integer <math>k </math> such that <math>P </math> is divisible by <math>3^k .</math> | ||
([[2006 AIME II Problems/Problem 3|Source]]) | ([[2006 AIME II Problems/Problem 3|Source]]) | ||
+ | |||
===Olympiad=== | ===Olympiad=== | ||
*Let <math>p_n (k) </math> be the number of permutations of the set <math>\{ 1, \ldots , n \} , \; n \ge 1 </math>, which have exactly <math>k </math> fixed points. Prove that <center><math>\sum_{k=0}^{n} k \cdot p_n (k) = n!</math>.</center> | *Let <math>p_n (k) </math> be the number of permutations of the set <math>\{ 1, \ldots , n \} , \; n \ge 1 </math>, which have exactly <math>k </math> fixed points. Prove that <center><math>\sum_{k=0}^{n} k \cdot p_n (k) = n!</math>.</center> | ||
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* A cool link to calculate factorials: http://www.nitrxgen.net/factorialcalc.php | * A cool link to calculate factorials: http://www.nitrxgen.net/factorialcalc.php | ||
+ | On that link, you can calculate factorials from <math>0!</math> to as much as <math>100000!</math> |
Latest revision as of 15:40, 17 March 2024
The factorial is an important function in combinatorics and analysis, used to determine the number of ways to arrange objects.
Contents
[hide]Factorials Video
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))
- (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 unitsdigit 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