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首页 Combinations and Permutations.pdf

Combinations and Permutations.pdf

Combinations and Permutations.p…

上传者: 搞搞陈 2013-03-13 评分 5 0 172 23 780 暂无简介 简介 举报

简介:本文档为《Combinations and Permutationspdf》,可适用于高等教育领域,主题内容包含MathPermutation•Anarrangementwhereorderisimportantiscalledapermutation•Ana符等。

MathPermutation•Anarrangementwhereorderisimportantiscalledapermutation•AnarrangementwhereorderisnotimportantiscalledcombinationSeatingArrangementPurpose:YouareaphotographersittingagroupinarowforpicturesYouneedtodeterminehowmanydifferentwaysyoucanseatthegroupPartOneTherearethreepeopletositdowninarowLetthecolorsred,blue,andgreenrepresentthethreepeopleIfRedisthefirsttositdown,showallthepossibleseatingarrangements(Usecoloredpencilstoshowthedifferentarrangements)IfBlueisthefirsttositdown,showallthesepossiblearrangementsIfGreenisthefirsttositdown,showallthesepossiblearrangementsHowmanytotalpossibleseatingarrangementsarethereforthreepeoplePartTwoTherearefivepeopleinthegroupLetthecolorsred,blue,green,yellow,andpurplerepresentthefivepeopleHowmanypeoplecouldsitdownfirstforthepictureIfRedsitsdown,howmanypeoplearelefttositdownIfBluesitsdown,howmanypeoplearelefttositdownIfGreensitsdown,howmanypeoplearelefttositdownIfYellowsitsdown,howmanypeoplearelefttositdownMultiplyhowmanycansitdownateachturnHowmanypossibleseatingarrangementsarethereforfivepeoplePartTwoTherearefivepeopleinthegroupLetthecolorsred,blue,green,yellow,andpurplerepresentthefivepeopleHowmanypeoplecouldsitdownfirstforthepictureIfRedsitsdown,howmanypeoplearelefttositdownIfBluesitsdown,howmanypeoplearelefttositdownIfGreensitsdown,howmanypeoplearelefttositdownIfYellowsitsdown,howmanypeoplearelefttositdownMultiplyhowmanycansitdownateachturnHowmanypossibleseatingarrangementsarethereforfivepeopleP=xxxx=ways•Aseatingarrangementisanexampleofapermutationbecausethearrangementofthe“n”objectsisinaspecificorderTheorderisimportantforapermutation•Whentheorderdoesnotmatter,itisacombination,becauseyouareonlyinterestedinthegroupExtend:Twelvepeopleneedtobephotographed,butthereareonlyfivechairs(Therestofthepeoplewillbestandingbehindandtheirorderdoesnotmatter)HowmanywayscanyousitthetwelvepeopleonthefivechairsExtend:Twelvepeopleneedtobephotographed,butthereareonlyfivechairs(Therestofthepeoplewillbestandingbehindandtheirorderdoesnotmatter)HowmanywayscanyousitthetwelvepeopleonthefivechairsExtend:Twelvepeopleneedtobephotographed,butthereareonlyfivechairs(Therestofthepeoplewillbestandingbehindandtheirorderdoesnotmatter)Howmanywayscanyousitthetwelvepeopleonthefivechairsxxxx=waysPermutation•Anarrangementwhereorderisimportantiscalledapermutation•Example:Mario,Sandy,Fred,andShannaarerunningfortheofficesofpresident,secretaryandtreasurerInhowmanywayscantheseofficesbefilledxx=TheofficescanbefilledwaysCombination•Anarrangementwhereorderisnotimportantiscalledcombination•Example:CharleshasfourcoinsinhispocketandpullsoutthreeatonetimeHowmanydifferentamountscanheget!xxxxPC•Determineifthesituationrepresentsapermutationoracombination:InhowmanywayscanfivebooksbearrangedonabookshelfinthelibraryInhowmanywayscanthreestudentcouncilmembersbeelectedfromfivecandidatesSevenstudentslineuptosharpentheirpencilsADJwillplaythreeCDchoicesfromtherequests•Determineifthesituationrepresentsapermutationoracombination:InhowmanywayscanfivebooksbearrangedonabookshelfinthelibrarypermutationInhowmanywayscanthreestudentcouncilmembersbeelectedfromfivecandidatescombinationSevenstudentslineuptosharpentheirpencilspermutationADJwillplaythreeCDchoicesfromtherequestscombination•Findthenumberofevents:InhowmanywayscanfivebooksbearrangedonabookshelfinthelibraryP=!=xxxx=ways•Findthenumberofevents:Inhowmanywayscanfivebooksbearrangedonabookshelfinthelibrary=waysInhowmanywayscanthreestudentcouncilmembersbeelectedfromfivecandidatesways!xxC•Findthenumberofevents:SevenstudentslineuptosharpentheirpencilsP=!=xxxxxx=waysADJwillplaythreeCDchoicesfromtherequests•Findthenumberofevents:SevenstudentslineuptosharpentheirpencilsP=!=xxxxxx=waysADJwillplaythreeCDchoicesfromtherequestsways!xxCFundamentalCountingPrincipleIfActcanbeperformedinmways,andActcanbeperformedinnwaysnomatterhowActturnsout,thenthesequenceActandActcanbeperformedinmnways•Example:EighthorsesAlabaster,Beauty,Candy,Doughty,Excellente,Friday,GreatOne,andHigh'nMightyrunarace•Inhowmanywayscanthefirstthreefinishersturnout•Example:EighthorsesAlabaster,Beauty,Candy,Doughty,Excellente,Friday,GreatOne,andHigh'nMightyrunarace•Inhowmanywayscanthefirstthreefinishersturnout•Example:EighthorsesAlabaster,Beauty,Candy,Doughty,Excellente,Friday,GreatOne,andHigh'nMightyrunarace•Inhowmanywayscanthefirstthreefinishersturnout•Example:EighthorsesAlabaster,Beauty,Candy,Doughty,Excellente,Friday,GreatOne,andHigh'nMightyrunarace•Inhowmanywayscanthefirstthreefinishersturnoutx•Example:EighthorsesAlabaster,Beauty,Candy,Doughty,Excellente,Friday,GreatOne,andHigh'nMightyrunarace•Inhowmanywayscanthefirstthreefinishersturnoutxx•Example:EighthorsesAlabaster,Beauty,Candy,Doughty,Excellente,Friday,GreatOne,andHigh'nMightyrunarace•Inhowmanywayscanthefirstthreefinishersturnoutxx=ways•Solution:ExtendingtheFundamentalCountingPrincipletothreeacts,finishingfirst("Act")canhappeninways,therearethenwaysinwhichfinishingsecond("Act")canoccur,andfinallytherearewaysinwhichthirdplace("Act")canbefilled,sothefirstthreefinisherscouldoccurin=ways•Example:Howmanywayscantossesofacointurnout•Solution:Eachoftheactscanoccurintwoways(HorT)Sothereare=differentsequencespossible•Example:Givenalistofblanks,inhowmanydifferentwayscanA,B,andCbeplacedintothreeoftheblanks,onelettertoablank(Twoblankswillbeempty)•Solution:TherearechoicesofablankforA,thenforB,andfinallyforCSothereare=waysinwhichthethreeletterscanbeplacedinthefiveblanks•OrdermattersinspellingandnumbersRATandTARaredifferentordersofthelettersA,R,andT,andcertainlyhavedifferentmeanings,asdoandThesearepermutations•Butmanytimesorderisnotimportant,thesearecombinationsABC,ACB,BAC,BCA,CAB,andCBAaresixdifferentpermutationsofthelettersA,B,andCfromthealphabet,buttheyrepresentjustonecombination•Chooseonecombinationoffourdifferentlettersfromthealphabet•Howmanypermutationsdoesthisonecombinationgive•Example:Inhowmanydifferentwayscouldacommitteeofpeoplebechosenfromaclassofstudents•positionstobefilled•peopletochosenfrom•lefttochoosefrom,etc•Thereforexxxx=,,permutations•Divideoutrepeatsof!•So,combinations•Example:Ifthefirstchosenwouldbechair,thenextonevicechair,thensecretary,andfinallytreasurer,inhowmanydifferentwayscouldacommitteeoffourbechosen•Example:Inatossofdifferenthonestcoins,whatistheprobabilityofgettingexactlyheads•Example:Intossesofa"loaded"coin,withprobabilityofheads=,whatistheprobabilityofgettingexactlyheads

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