Permutations, Combinations and the Binomial Theor Combinations and the Binomial Theorem ... How many ways can the letters in the word PENCIL ... In how many ways can 6 people be arranged

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  • Section9.1Permutations

    1

    CHAPTER9Permutations,CombinationsandtheBinomialTheorem

    (Chapter11inResource)

    Howmanywayscanitemsbearranged?

    FundamentalCountingPrinciple Factorial Permutation Combination

    CountingMethods

    Factorial

    multiplyconsecutivenumbersdecreasingby1.

    Permutation

    ordermatters(selectionofobjects)

    Combination

    orderdoesnotmatter(selectionofobjects)

    Examples: passwordorcode selectingagrouptobepresident,vicepresident,treasurer awardingmedalsto1stplace,2ndplace,3rdplace

    Examples: lottery pickateamof5peoplefromagroupof10

  • Section9.1Permutations

    2

    (1) HowmanywayscanyouarrangethelettersinthewordMICRO?

    (2) HowmanywayscanyouarrangethelettersinthewordPOSSIBILITY?

    (3) HowmanythreeletterwordscanbemadefromthelettersofthewordKEYBOARD?

    (4) HowmanywayscanBeth,John,Doug,andMarybeseatedinarowifDougmustbeinthesecondseat?

    (5) HowmanyarrangementsofthewordACTIVEarethereifCandEmustalwaysbetogether?

    ThinkAbout:

    arrangingasubsetofitems

    repetition

    specificposition

    itemstogether

    norepetition

  • Section9.1Permutations

    3

    FundamentalCountingPrinciple

    determinethetotalnumberofpossibleoutcomesthatcanoccurwithagroup(s).

    Example1

    Theschoolcafeteriaadvertisesthatitcanserveupto24differentmealsconsistingofoneitemfromeachofthethreecategories:

    Fruit: Apples(A),Bananas(B)orCantaloupe(C)

    Sandwiches: RoastBeef(R)orTurkey(T)

    Beverages: Lemonade(L),Milk(M),OrangeJuice(O)orPineappleJuice(P)

    Istheiradvertisingcorrect?

    Treediagram

    Ifonetaskcanbeperformedinways,

    asecondtaskcanbeperformedinways,

    andathirdtaskbeperformedinways,thenthenumberofwaysto

    performallthetaskstogetheris

    FundamentalCountingPrinciple:

    choicesforfruit

    choicesforsandwich

    choicesforbeverage

  • Section9.1Permutations

    4

    DistinguishbetweenthewordsAND/OR

    #waystochooseafruitandasandwichandabeverage,

    3fruitchoicesx2sandwichchoicesx4beveragechoices

    =24possibilities

    #waystochooseafruitorasandwichorabeverage

    3fruitchoices+2sandwichchoices+4beveragechoices

    =9possibilities

    (multiplyindividualselections)

    (addindividualselections)

  • Section9.1Permutations

    5

    Example2

    Astoremanagerhasselected4possibleapplicantsfortwodifferentpositionsatadepartmentstore.Inhowmanywayscanthemanagerfillthepositions?

    Example3

    HowmanywayscanthelettersinthewordPENCILbearranged?

    FundamentalCountingPrinciple

    #ofchoicesforposition1 #ofchoicesforposition2

    #ofwaystofillthepositions

    Idea:Wehave6objectsand6possiblepositionstooccupy

    and

    ArrangementsWithoutRestrictions

  • Section9.1Permutations

    6

    Factorial:

    FactorialManyexamplesinvolvearrangementswhereyoumultiplyconsecutivenumbersdecreasingby1.

    Previousexample:

    Thisproductcanbewrittenincompactformas

    General:where

    Example4:Simplifyfactorialexpressions

    a)

    b)

    c)

  • Section9.1Permutations

    7

    Example5

    Inhowmanywayscanateacherseat4boysandthreegirlsinarowof7seatsifaboymustbeseatedateachoftherow?

    Restriction:aboymustbeineachendseat.

    Fillseats1and7first Thenfillremainingseats

    FundamentalCountingPrinciple

    ArrangementsWithRestrictions

    Example6

    A5digitpasswordistobecreatedusingthedigits09.

    a)Howmanyarrangementsarepossibleifrepetitionisallowed?

    b)Howmanyarrangementsarepossibleifrepetitionisnotallowed?

    c)Howmanyarrangementsarepossibleifconsecutivedigitsarenotallowed?

  • Section9.1Permutations

    8

    Permutation

    arrangingobjectswhereorderisimportant

    Formulatodeterminethenumberofpermutationsofdifferentelementstakenatatime

    Example7

    Howmanywaysaretheretoarrange3peopleofagroupof5inaline?

    FundamentalCountingPrinciple:

    PermutationFormula:

    arrangingasubsetofitems

    onlysomeoftheitemsareusedinthearrangement

  • Section9.1Permutations

    9

    Example8

    Inhowmanywayscan6peoplebearrangedinalineforaphotograph?

    Example9

    Howmany4letterwordscanbecreatedifrepetitionsarenotallowed?

  • Section9.1Permutations

    10

    Example10

    Ifthereare7membersonthestudentcouncil,howmanywayscanthecouncilselect3studentstobethepresident,vicepresidentandthetreasurer?

    Example12

    AcodeconsistsofthreeletterschosenfromAtoZandthreedigitschosenfrom0to9,withnorepetitionsoflettersornumbers.Determinethetotalnumberofpossiblecodes.

    Example11

    HowmanythreeletterwordscanbemadefromthelettersofthewordKEYBOARD?

  • Section9.1Permutations

    11

    Example14:

    Showthatwithouttechnology.

    Example13:Solveequationsusingwhere

    a)Solve

    b)Solve

    c)Solve

  • Section9.1Permutations

    12

    Example15

    Ifthereare56gamesinaseries,andeachteamplayedeveryotherteamtwice,onceathomeandonceaway,howmanyteamsarethere?

  • Section9.1Permutations

    13

    PermutationswithRepeatingObjects

    Example16

    HowmanydifferentwayscanyouarrangethelettersinthewordPOOL?

    Example17

    HowmanydifferentwayscanyouarrangethelettersinthewordMATHEMATICS?

    Insomecases,someoftheitemswewanttoarrangeareidentical.

    Ifasetofobjects,withofonekindthatareidentical,ofasecondkind.....theentiresetofobjectscanbearrangedin

  • Section9.1Permutations

    14

    WithConstraints

    twoormoreobjectsmustbeplacedtogether twoormoreobjectscannotbeplacedtogether certainobjectsmustbeplacedincertainpositions

    Example18

    Howmanywayscanagroupof5peoplebearrangedinalineiftwoofthemaregoodfriendsandwanttosittogether.

    Permutations

    whencertainitemsaretobekepttogether,treatthejoineditemasiftheywereonlyoneobject.

    Mary Beth

    Dave Kyle

    John

  • Section9.1Permutations

    15

    Example19

    Howmanywayscanagroupof5peoplebearrangedinalineifMaryandBethshouldnotsittogether.

    total#ofarrangementswithnorestrictions

    arrangementswithMaryandBethtogether

    Complement

  • Section9.1Permutations

    16

    ArrangementsinvolvingCases

    Someproblemshavemorethanonecase.Calculatethenumberofarrangementsforeachcaseandthenaddupthevaluesforallcasestoobtainthetotal.

    Example20

    Determinethenumberofarrangementsof4girlsand3boysinarowofsevenseatsiftheendsoftherowsmustbeeitherbothfemaleorbothmale.

    KeyWords:AtLeast,AtMost,Either

    CaseforFemales CaseforMales4girls,3boys

    Page 1: May 9-8:58 AMPage 2: May 6-7:20 PMPage 3: May 4-10:26 AMPage 4: May 4-4:33 PMPage 5: May 4-10:42 AMPage 6: May 4-11:05 AMPage 7: May 4-11:12 AMPage 8: May 4-11:27 AMPage 9: May 9-10:01 AMPage 10: May 4-11:35 AMPage 11: May 4-12:13 PMPage 12: May 9-9:56 AMPage 13: May 4-1:18 PMPage 14: May 4-11:47 AMPage 15: May 6-8:58 PMPage 16: May 4-12:13 PM

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