combination-and-permutation

COMBINATIONANDPERMUTATIONFORMULATHEMULTIPLICATIONPRINCIPLEE.g.1CountingPathsthroughaMazeAmedicalresearcherwishestotesttheeffectofadrugonarat’sperceptionbystudyingtherat’sabilitytorunamazewhileundertheinfluenceofthedrug.ThemazeisconstructedsothattoarriveattheexitpointC,theratmustpassthroughacentralpointB.TherearefivepathsfromtheentrypointAtoB,andthreepathsfromBtoC.InhowmanydifferentwayscantheratruntothemazefromAtoC?MultiplicationPrincipleSupposethatataskiscomposedoftwoconsecutivechoices.Ifchoice1canbeperformedinmwaysand,foreachofthese,choice2canbeperformedinnways,thenthecompletetaskcanbeperformedinm∙nways.E.g.2CountingRoutsforaTripAnairlinepassengermustflyfromNewYorktoFrankfurtviaLondon.Thereare8flightsleavingNewYorkforLondon.Alloftheseprovideconnectionsonanyoneof19flightsfromLondontoFrankfurt.Inhowmanydifferentwayscanthepassengerbookreservations?E.g.3Acorporationhasaboardofdirectorsconsistingof10members.Theboardmustselectfromitsmembersachairperson,vicechairperson,andsecretary.Inhowmanywayscanthisbedone?E.g.4Inhowmanywayscanabaseballteamofnineplayersarrangethemselvesinalineforagrouppicture?E.g.5Acertainstateusesautomobilelicenseplatesthatconsistofthreelettersfollowedbythreedigits.Howmanysuchlicenseplatesarethere?ProblemAHowmanywords(bywhichwemeanstringsofletters)oftwodistinctletterscanbeformedfromtheletters{a,b,c}?ProblemBAconstructioncrewhasthreemembers.Ateamoftwomustbechosenforaparticularjob.Inhowmanywayscantheteambechosen?P(n,r)=thenumberofpermutationsofnobjectstakenratatimeC(n,r)=thenumberofcombinationofnobjectstakenratatime!)(!xxnnxnCxn)!(!xnnPxnP(n,1)=nP(n,1)=n(n-1)P(n,1)=n(n-1)(n-2)P(n,r)=n(n-1)(n-2)……(n-r+1)1)1()1()1(!),(),(rrrnnnrrnPrnCApplyingthePermutationandCombinationFormula(a)P(100,2)(b)P(6,4)(c)P(5,5)(d)C(100,2)(e)C(6,4)(f)C(5,5)E.g.6Theboardofdirectorsofacorporationhas10members.Inhowmanywayscantheychooseacommitteeofthreeboardmemberstonegotiateamerger?E.g.7Eighthorsesareenteredinaraceinwhichafirst,secondandthirdprizewillbeawarded.Assumingnoties,howmanydifferentoutcomesarepossible?E.g.8Apoliticalpollsterwishedtosurvey1500individualschosenfromasampleof5,000,000adults.Inhowmanywayscanthe1500individualsbechosen?E.g.9Aclubhas10members.Inhowmanywayscantheychooseaslateoffourofficers,consistingofapresident,vicepresident,secretary,andtreasurer?E.g.10Threecouplesgoonamoviedate.Inhowmanywayscantheybeseatedinarowofsixseatssothateachcoupleisseatedtogether?E.g.11Ifyouhavesixbooks,inhowmanywayscanyouselectfourbooksandarrangethemonashelf?E.g.12CommitteeSelectionThestudentcouncilatCityCollegehas12students,ofwhich3areseniors.Howmanycommitteesof4studentscanbeselectedifatleastonememberofeachcommitteemustbeasenior?C(12,4)–C(9,4)E.g.13GroupPictureThestudentcouncilatGothamCollegeismadeupoffourfreshmen,fivesophomores,sixjuniors,andsevenseniors.Ayearbookphotographerwouldliketolineupthreecouncilmembersfromeachclassforapicture.Howmanydifferentpicturesarepossibleifeachgroupofclassmatesstandstogether?4!∙P(4,3)∙P(5,3)∙P(6,3)∙P(7,3)E.g.14DisplayingPaintingsAnartgalleryhasfivepaintings,byeachofthreeartists,hanginginarow,withpaintingsbythesameartistgroupedtogether.Howmanydifferentarrangementsarepossible?3!∙5!3LetSbeasamplespaceconsistingofNequallylikelyoutcomes.LetEbeanyevent.ThenNEE]inoutcomesofnumber[)Pr(E.g.15SelectingBallsfromanUrnAnurncontainseightwhiteballsandtwogreenballs.Asampleofthreeballsisselectedatrandom.Whatistheprobabilityofselectingonlywhiteballs?NEE]inoutcomesofnumber[)Pr(12056)3,10()3,8(CCExercise:BallsinanUrnAnurncontainssevengreenballsandfivewhiteballs.Asampleofthreeballsisselectedatrandomfromtheurn.Findtheprobabilitythat(a)Onlygreenballsareselected(b)Atleastonewhiteballisselected.E=“allthreeballsselectedaregreen”F=“atleastonewhiteballisselected”TheeventEisthecomplementofFE=FCPr(F)=1-Pr(FC)=1-Pr(E)Exercise:BirthdayThreepeoplearechosenatrandom.Whatistheprobabilitythatatleasetwoofthemwerebornonthesamedayoftheweek?Exercise:randomgradingTheinstructorinawritingseminardoesnothavetimetogradetheeightessayssubmittedbyhisstudents.Hedecidestousearandom-numbergeneratortoassigneachessayagradefrom81through100,inclusive.Whatistheprobabilitythatatleasttwostudentswillreceivethesamegrade?