combination with repetition examples with solutions pdf

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Wemustchoosethosefourelementsfromtheset{6,7,,}nocombinationsareoverlooked–Nowweneedtoplacethen0’sintothekrunsIfX={x1,x2,,we writeanr-combinationwithrepetitionallowed,ormultisetof1)+1))+(k(k;(k+1)1)=C(m+1;k).IAnotherwaytoseethis:Computetotalofpermutations(n!) andthendividebyofrelativeorderingsbetweenobjectsoftype(n1!),ofrelativeorderingsofobjectsoftype(n2!)etcn1!n2!nk!But,wehavearestrictionhere tooEachrunmusthaveatleastonesowehavenk0stoplaceInfiniterepetitionnumbers:examplesExampleCountthenumberofintegersolutionstothe equationx1++x7=;suchthat(a)x1;;x;(b)x1;;x;(c)x,x,x3;;xIneachcase,weneedtocountthenumberofvectors(x1;;x7)thatsatisfythegivenconditions CombinationsProblemExampleLetS={1,2,3,,}Hint:Whatisnandwhatisr?Givenrobjectsandntypesofobjectstochoosefrom,thenumberofselections withrepetitionsis:r+(n1)QHowmanywaysaretheretodistributeidenticalchocolatebarsandidenticalsticksofgumtochildren?Let'slookatasimple example:Provethatforallpositiveintegersk,()!Inessence,weknowthatmustbeoneofourelements,sowearereallyfreetochooseonlyelements. CombinationswithRepetitionExample:Howmanywaysaretheretoselectfivebillsfromaboxcontainingatleastfiveofeachofthefollowingdenominations:$1, $2,$5,$10,$20,$50,and$?HowmanysubsetsAofScontainelements,withbeingtheleast?Orderdoesnotmatter,repeatedobjectsThisisktypeswithnk CombinationsWithRepetitionCombinationsrepresentthenumberofdifferentpossiblewayswecanpickanumberofelementsSolution[1,];[];[]all combinationswith1,[1,2,2];[1,2,3];[1,2,4];alladditionalcombinationswith1,[1,3,3];[1,3,4];[1,4,4];alladditionalcombinationswith1,or1,[2,2,2];[2,2,3]; [2,2,4];alladditionalcombinationswith2,[2,3,3];[2,3,4];[2,4,4];alladditionalcombinationswith2,3CombinationswithRepetitionConsidercountingthe numberofpermutationsofthensymbolsx1,x1,x2,x2,xk,XofnelementsisanunorderedselectionofelementstakenfromXwithrepetitionallowed Solution:EachIThissimpliesto:n!Theorem:Thenumberofr-combinationsfromasetwithnelementswhenrepetitionofelementsisallowedisInspecialletus exploreaspecificexampleCombinationswithRepetitionExample:Howmanywaysaretheretoselectfivebillsfromaboxcontainingatleastfiveofeachofthe followingdenominations:$1,$2,$5,$10,CombinationswithRepetitionInstructor:IslDillig,CSH:DiscreteMathematicsCombinatorics/Example1Tryit! Example:HowmanywaystoselectbillsCombinationswithRepetitionExample:Howmanysolutionsdoestheequationx+x+x=have,wherex1,xandxare nonnegativeintegers?Theyallboildowntothequestion:howmanywaysaretheretoselectkelementsfromCombinationswithRepetitionTheorem:Thereare C(n+r¡1;r)r-combinationsfromasetwithnelementswhenrepetitionisallowedThere'sabitmorevarietywiththesetypesofproblemsisaninteger Considerthissolution(3,8,2,6,1,1,4,5)Thisisinonetoonecorrespondencewithx1=3,x2=8,x3=2,x4=6,x5=1,x6=1,andx7=Similarly,givenany solutiontotheinequality,wecancalculatetheuniquecorrectsolutionfortheequalitybelowitHowmanywayscanyoubuyfruitifyouroptionsareapples, bananas,pears,andoranges?choicesfromoptionswithrepetition,sothenumberofwaysis+==Howmanywayscanyougivecookiestofriendsifeachfriendgets atleastcookie?