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>10000!</math> is 38660 digits long and has 2499 terminal zeroes! | ||
* <math>100000!</math> is 456574 digits long and has 24999 terminal zeroes! | * <math>100000!</math> is 456574 digits long and has 24999 terminal zeroes! | ||
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== Additional Information == | == Additional Information == | ||
− | By [[mathematical convention|convention]] and rules of an | + | 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> | + | On that link, you can calculate factorials from <math>0!</math> to as much as <math>100000!</math> |
Latest revision as of 16:40, 17 March 2024
The factorial is an important function in combinatorics and analysis, used to determine the number of ways to arrange objects.
Contents
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