Principles-of-Math-12----Permutations-and-Combinat

PrinciplesofMathematics12:Explained!!:TheFundamentalCountingPrincipleTheFundamentalCountingPrinciple:Thisisaneasywaytodeterminehowmanywaysyoucanarrangeitems.Thefollowingexamplesillustratehowtouseit:Example1:HowmanywayscanyouarrangethelettersinthewordMICRO?Example2:Howmanywayscan8differentalbumsbearranged?Wecouldapproachthisquestioninthesamewayasthelastonebyusingthespacesandmultiplyingallthenumbers,butthereisashorterway.Thefactorialfunctiononyourcalculatorwillperformthiscalculationforyou!8!=8•7•6….2•18!=40320Questions:1)HowmanywayscanthelettersinthewordPENCILbearranged?2)Iftherearefourdifferenttypesofcookies,howmanywayscanyoueatallofthem?3)Ifthreealbumsareplacedinamulti-discstereo,howmanywayscanthealbumsbeplayed?4)Howmanywayscanyouarrangeallthelettersinthealphabet?5)Howmanywayscanyouarrangethenumbers24through28(inclusive)?Answers:1)6!=7202)4!=243)3!=64)26!=4.03•10265)5!=120TI-83InfoYoucangetthefactorialfunctionusing:MathÆPrbÆ!Thebasicideaiswehave5objects,and5possiblepositionstheycanoccupy.PrinciplesofMathematics12:Explained!:RepetitionsNotAllowedRepetitionsNotAllowed:Inmanycases,someoftheitemswewanttoarrangeareidentical.Forexample,inthewordTOOTH,ifweexchangetheplacesofthetwoO’s,westillgetTOOTH.Becauseofthis,wehavetogetridofextraneouscasesbydividingoutrepetitions.Example1:HowmanywayscanyouarrangethelettersinthewordTHESE?Dothisasafraction.Factorialthetotalnumberoflettersandputthisontop.Factorialtherepeatedlettersandputthemonthebottom.Example2:HowmanywayscanyouarrangethelettersinthewordREFERENCE?5!120==2!260Questions:1)HowmanywayscanthelettersinthewordSASKATOONbearranged?2)HowmanywayscanthelettersinthewordMISSISSIPPIbearranged?3)HowmanywayscanthelettersinthewordMATHEMATICSbearranged?4)Ifthereareeightcookies(4chocolatechip,2oatmeal,and2chocolate)inhowmanydifferentorderscanyoueatallofthem?5)Ifamultiplechoicetesthas10questions,ofwhichoneisansweredA,4areansweredB,3areansweredC,and2areansweredD,howmanyanswersheetsarepossible?9!362880==2!•4!2•247560TI-83InfoMakesureyouputthedenominatorinbracketsoryou’llgetthewronganswer!Answers:9!1)453602!2!2!11!2)346504!4!2!11!3)49896002!2!2!8!4)4204!2!2!10!5)126004!3!2!=••=••=••=••=••PrinciplesofMathematics12:Explained!:RepetitionsAreAllowedRepetitionsAreAllowed:Sometimesweareinterestedinarrangementsallowingtheuseofitemsmorethanonce.Example1:Thereare9switchesonafusebox.Howmanydifferentarrangementsarethere?Example2:Howmany3letterwordscanbecreated,ifrepetitionsareallowed?Questions:1)Ifthereare4lightswitchesonanelectricalpanel,howmanydifferentordersofon/offarethere?2)Howmany5letterwordscanbeformed,ifrepetitionsareallowed?3)Howmanythreedigitnumberscanbeformed?(Zerocan’tbethefirstdigit)4)Acoathangerhasfourknobs.Ifyouhave6differentcolorsofpaintavailable,howmanydifferentwayscanyoupainttheknobs?Therearetendigitsintotal,fromzerotonine.Answers:1)24=162)265=118813763)9•10•10=9004)64=1296Thereare26letterstochoosefrom,andweareallowedtohaverepetitions.Thereare263=17576possiblethreeletterwords.Eachswitchhastwopossiblepositions,onoroff.Placinga2ineachofthe9positions,wehave29=512.PrinciplesofMathematics12:Explained!:ArrangingASubsetArrangingasubsetofitems:Sometimesyouwillbegivenabunchofobjects,andyouwanttoarrangeonlyafewofthem:Example1:Thereare10peopleinacompetition.Howmanywayscanthetopthreebeordered?Example2:Thereare12moviesplayingatatheater,inhowmanywayscanyouseetwoofthemconsecutively?Youcouldusethespaces,butlet’strythisquestionwiththepermutationfeature.Thereare12movies,andyouwanttosee2,sotype12P2intoyourcalculator,andyou’llget132.Example3:Howmany4letterwordscanbecreatedifrepetitionsarenotallowerd?Theansweris:26P4=358800Questions:1)HowmanythreeletterwordscanbemadefromthelettersofthewordKEYBOARD2)Ifthereare35songsandyouwanttomakeamixCDwith17songs,howmanydifferentwayscouldyouarrangethem?3)Therearesixdifferentcoloredballsinabox,andyoupullthemoutoneatatime.Howmanydifferentwayscanyoupulloutfourballs?4)Acommitteeistobeformedwithapresident,avice-president,andatreasurer.Thereare10peopletobeselectedfrom.Howmanydifferentcommitteesarepossible?5)Abaseballleaguehas13teams,andeachteamplayseachothertwice;onceathome,andonceaway.Howmanygamesarescheduled?Answers:1)8P3=3362)35P17=1.6•10243)6P4=3604)10P3=7205)13P2=156ORORAfastwaytodoquestionslikethisistousethenPrfeatureonyourcalculator.nisthetotalnumberofitems.risthenumberofitemsyouwanttoorder.ForExample1,youwouldtype:10ÆMathÆPRBÆnPrÆenterÆ3Sinceonly3positionscanbefilled,wehave3spaces.Multiplying,weget720.PrinciplesofMathematics12:Explained!:SpecificPositionsSpecificPositions:Frequentlywhenarrangingitems,aparticularpositionmustbeoccupiedbyaparticularitem.Theeasiestwaytoapproachthesequestionsisbyanalyzinghowmanypossiblewayseachspace