6. Enumerating Anagrams / Computing Factorials

16 Pages·2011·0.27 MB·English

6. Enumerating Anagrams / Computing Factorials
Robert Snapp
[email protected]
DepartmentofComputerScience
UniversityofVermont
RobertR.Snapp©2011 6.EnumeratingAnagrams/ComputingFactorials CS21,Fall2011 1/13
1 Enumeratinganagrams
2 ComputingfactorialswithDrRacket
3 Astronomicalnumbers
RobertR.Snapp©2011 6.EnumeratingAnagrams/ComputingFactorials CS21,Fall2011 2/13
Anagrams: Review
InCS32,thewordanagramsignifiesapermutation,orrearrangementofagroupof
lettersorsymbols. ThusthelettersinthewordCAThassixdifferentanagrams:
ACT ATC
CAT CTA
TAC TCA
(NotethatonlytwooftheseareEnglishwords,theremainingfourarejibberish,but
areneverthelessanagrams,byourdefinition.)
Thenumberofdifferentanagramsofawordhavingndifferentlettersequals
nŠDn(cid:2).n(cid:0)1/(cid:2).n(cid:0)2/(cid:2)(cid:1)(cid:1)(cid:1)(cid:2)3(cid:2)2(cid:2)1:
ThenotationnŠiscalled“n-factorial.”
RobertR.Snapp©2011 6.EnumeratingAnagrams/ComputingFactorials CS21,Fall2011 3/13
Anagrams: Review
Iftheinitialwordhasrepeatedletters,thenthenumberofanagramsisreduced. Thus
thewordMALLhas12anagrams:
ALLM ALML AMLL
LALM LAML LLAM LLMA LMAL LMLA
MALL MLAL MLLA
Withfourletters,wemightexpecttofind4Špermuations. Howeverthisnumber
overcountsthecorrectvalue,becauseLLhasonlyoneanagram,nottwo. Thuswe
needtodivide4Šby2Š. Thus
4Š 24
D D12:
2Š 2
RobertR.Snapp©2011 6.EnumeratingAnagrams/ComputingFactorials CS21,Fall2011 4/13
Anagrams: Review
HowmanyanagramsareinthewordLULL?
HowmanyareinTOOT?
RobertR.Snapp©2011 6.EnumeratingAnagrams/ComputingFactorials CS21,Fall2011 5/13
Anagrams: Review
Recall,thatawordconsistingofk differentletters,withr repetitionsofthefirst,r of
1 2
thesecond,:::,andr ofthek-thletter,has
k
.r Cr C(cid:1)(cid:1)(cid:1)Cr /Š
1 2 k
:
r Š r Š (cid:1)(cid:1)(cid:1) r Š
1 2 k
uniqueanagrams. Thusthe11letterwordABRACADABRA,whichhas5As,2Bs,
1C,1D,and2Rs,has
11Š 11(cid:2)10(cid:2)9(cid:2)8(cid:2)7(cid:2)6(cid:2)5Š
D (Cancelcommonfactors!)
5Š (cid:1)2Š (cid:1)1Š (cid:1)1Š (cid:1)2Š 5Š(cid:1)2(cid:1)2
D11(cid:2)10(cid:2)9(cid:2)2(cid:2)7(cid:2)6
D99(cid:2)840D83;160
uniqueanagrams. (Wheneveronehasafractionwithfactorialsappearinginboththe
numeratoranddenominator,onecanalwayscancelcommonfactors.)
RobertR.Snapp©2011 6.EnumeratingAnagrams/ComputingFactorials CS21,Fall2011 6/13
Computing Factorials
Computinglargefactorialsthoughisdifficultandtimeconsuming,evenwitha
calculator. FortunatelywecanletDrRacketdothehardworkforus. Firstthough
let’sexaminethemathematicaldefinitionofthefactorial: Foranynaturalnumber1 n
(
1; ifnD0;
nŠD
n(cid:2).n(cid:0)1/Š; otherwise.
Thestatementthat0ŠD1isausefulconvention. Itmightsoundoddthatthereis
exactlyonewaytopermuteananagramthathaszeroletters. However,0ŠD10,as
bothareproductsthatcontainzerofactors. Andsince,10 D1,itfollowsthat0ŠD1.
ThestatementthatnŠDn(cid:2).n(cid:0)1/Š,followsfromobservingthatinsideevery
factorial,e.g.,5ŠD5(cid:2)4(cid:2)3(cid:2)2(cid:2)1,thereisaslightlysmallerone: Since
4ŠD4(cid:2)3(cid:2)2(cid:2)1,wehave5ŠD5(cid:2)4Š.
1Anaturalnumberisanon-negativeinteger
RobertR.Snapp©2011 6.EnumeratingAnagrams/ComputingFactorials CS21,Fall2011 7/13
Computing factorials in DrRacket
Inthetop(Definition)window,enterthefollowingfunctiondefinition.
(define (factorial n)
( if (= n 0)
1
((cid:3) n (factorial ((cid:0) n 1)))))
Savethedefinitioninafilecalledfactorial.rkt. Thespecialform“define”enables
ustodefinenewfunctionsinRacket. Don’tworrythatyoudon’tunderstandthe
syntaxrightaway;ittakessomegettingusedto. Canyouobserveanysimilarity
betweenthisfunction,andthemathematicaldefinitiononthepreviousslide?
Pressthe“Run”button(withthegreenrunningstickfigure)ontheupperright. Then
youcanevaluateexpressionslike
(factorial 7) or (factorial 52)
inthelower“Interaction”window. Justtypeinanexpressionafterthecommand
prompt“>”andpressthereturnkeyonthekeyboard.
RobertR.Snapp©2011 6.EnumeratingAnagrams/ComputingFactorials CS21,Fall2011 8/13
DrRacket on a Macintosh
RobertR.Snapp©2011 6.EnumeratingAnagrams/ComputingFactorials CS21,Fall2011 9/13
Other ways to express the factorial
Thepreviousdefinitionoffactorialisconcise. Theindentationsandlinebreaksare
forourbenefit: DrRacketignoresthem. Wecanalsoaddcomments(whichbegin
withasemicolon)tohelpremindushowthisfunctionworks. Wecanalsointroduce
otherhelperfunctionsifwelike:
;;; dec stands for decrement; it takes one away
;;; from its argument. So, (dec 3) => 2.
(define (dec n)
((cid:0) n 1))
;;; ! returns the factorial of its argument, so (! 7) => 5040.
(define (! n)
( if (zero? n)
1 ; 0! = 1, otherwise
((cid:3) n (! (dec n))))) ; n! = n (cid:3) (n(cid:0)1)!
Thefunctionzero? isalreadypartofracket: (zero? 0)=>#t,i.e.. “true,”and(zero? 1)
=>#f,thatis,“false.”
TrycopyingtheaboveintotheDefinitionwindowinDrRacket,andrunafewtests.
RobertR.Snapp©2011 6.EnumeratingAnagrams/ComputingFactorials CS21,Fall2011 10/13