Probability Day 3 - Permutations and Combinations

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Text of Probability Day 3 - Permutations and Combinations

  • FactorialNotation Theexpression654321=canbe writtenas6!,whichisreadassixfactorial. Ingeneral,n!istheproductofallthecounting numbersbeginningwithnandcountingbackwards to1. Wedefine0!tobe1. FactorialonyourTIcalculator.
  • Example: Findthevalueofeachexpression: a)3! b)0! c)3!+2! d)
  • FundamentalCountingPrinciple: Ifoneactivitycanoccurinanyofmwaysand, followingthis,asecondactivitycanoccurinany ofnways,thenbothactivitiescanoccurinthe ordergiveninm*nways.
  • PermutationFormulaanarrangementofobjectsinsomespecificorder IngeneralP(n,r)meansthenumberofpermutations ofnitemsarrangedratatime. Theformulaforpermutationis PermutationsonyourTIcalculator Note: nPn=n! 3P3=3!=3*2*1=6
  • Wordsusedinpermutationproblems: arrangement lineup president,vicepresident,secretary 1st,2nd,3rdplace Example: Alicenseplatebeginswiththreeletters.IfthepossiblelettersareA,B,C,Dand E,howmanydifferentarrangementsoftheseletterscanbemadeifnoletteris usedmorethanonce?
  • Permutationswithrepetition Ifwewanttoarrangeitemswhentherearemorethan oneofthesameitem,weneedtodividebythenumber ofidenticalitems: Example: Findthenumberofarrangementsofthelettersthatcan beformedfromthelettersIDENTITY,usingeachletter Solution: Example:Findthenumberofarrangementsof lettersthatcanbeformedfromtheletters: MINIMUM Solution:
  • Combinations Anarrangementofobjectsinwhichtheorderis notimportantiscalledacombination.Thisis differentfrompermutationwheretheorder matters.Forexample,supposewearearranging thelettersA,BandC.Inapermutation,the arrangementABCandACBaredifferent.But,ina combination,thearrangementsABCandACBare thesamebecausetheorderisnotimportant.
  • Thenumberofcombinationsofnthingstakenrat atimeiswrittenasC(n,r). Theformulaisgivenby: CombinationsonyourTIcalculator
  • Wordsusedincombinationproblems: committee group team Example: Inhowmanywayscanacoachchoosethree swimmersfromamongfiveswimmers?
  • Example6: Thereare5redand4whitemarblesinanurn.Amarbleisdrawnfromtheurnandnot replaced.Then,asecondmarbleisdrawn. a.Inhowmanywayscanaredmarbleandawhitemarblebedrawninthatorder? b.Inhowmanywayscanaredmarbleandawhitemarblebedrawnineitherorder? Example7: Anurncontainsthreewhiteballsandfourredballs.Twoballsarechosenat random.Howmanywayscanyouchoseatleastoneoftheredballs?