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
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(1) HowmanywayscanyouarrangethelettersinthewordMICRO?
(2) HowmanywayscanyouarrangethelettersinthewordPOSSIBILITY?
(3) HowmanythreeletterwordscanbemadefromthelettersofthewordKEYBOARD?
(4) HowmanywayscanBeth,John,Doug,andMarybeseatedinarowifDougmustbeinthesecondseat?
(5) HowmanyarrangementsofthewordACTIVEarethereifCandEmustalwaysbetogether?
ThinkAbout:
arrangingasubsetofitems
repetition
specificposition
itemstogether
norepetition
Section9.1Permutations
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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
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DistinguishbetweenthewordsAND/OR
#waystochooseafruitandasandwichandabeverage,
3fruitchoicesx2sandwichchoicesx4beveragechoices
=24possibilities
#waystochooseafruitorasandwichorabeverage
3fruitchoices+2sandwichchoices+4beveragechoices
=9possibilities
(multiplyindividualselections)
(addindividualselections)
Section9.1Permutations
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Example2
Astoremanagerhasselected4possibleapplicantsfortwodifferentpositionsatadepartmentstore.Inhowmanywayscanthemanagerfillthepositions?
Example3
HowmanywayscanthelettersinthewordPENCILbearranged?
FundamentalCountingPrinciple
#ofchoicesforposition1 #ofchoicesforposition2
#ofwaystofillthepositions
Idea:Wehave6objectsand6possiblepositionstooccupy
and
ArrangementsWithoutRestrictions
Section9.1Permutations
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Factorial:
FactorialManyexamplesinvolvearrangementswhereyoumultiplyconsecutivenumbersdecreasingby1.
Previousexample:
Thisproductcanbewrittenincompactformas
General:where
Example4:Simplifyfactorialexpressions
a)
b)
c)
Section9.1Permutations
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Example5
Inhowmanywayscanateacherseat4boysandthreegirlsinarowof7seatsifaboymustbeseatedateachoftherow?
Restriction:aboymustbeineachendseat.
Fillseats1and7first Thenfillremainingseats
FundamentalCountingPrinciple
ArrangementsWithRestrictions
Example6
A5digitpasswordistobecreatedusingthedigits09.
a)Howmanyarrangementsarepossibleifrepetitionisallowed?
b)Howmanyarrangementsarepossibleifrepetitionisnotallowed?
c)Howmanyarrangementsarepossibleifconsecutivedigitsarenotallowed?
Section9.1Permutations
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Permutation
arrangingobjectswhereorderisimportant
Formulatodeterminethenumberofpermutationsofdifferentelementstakenatatime
Example7
Howmanywaysaretheretoarrange3peopleofagroupof5inaline?
FundamentalCountingPrinciple:
PermutationFormula:
arrangingasubsetofitems
onlysomeoftheitemsareusedinthearrangement
Section9.1Permutations
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Example8
Inhowmanywayscan6peoplebearrangedinalineforaphotograph?
Example9
Howmany4letterwordscanbecreatedifrepetitionsarenotallowed?
Section9.1Permutations
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Example10
Ifthereare7membersonthestudentcouncil,howmanywayscanthecouncilselect3studentstobethepresident,vicepresidentandthetreasurer?
Example12
AcodeconsistsofthreeletterschosenfromAtoZandthreedigitschosenfrom0to9,withnorepetitionsoflettersornumbers.Determinethetotalnumberofpossiblecodes.
Example11
HowmanythreeletterwordscanbemadefromthelettersofthewordKEYBOARD?
Section9.1Permutations
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Example14:
Showthatwithouttechnology.
Example13:Solveequationsusingwhere
a)Solve
b)Solve
c)Solve
Section9.1Permutations
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Example15
Ifthereare56gamesinaseries,andeachteamplayedeveryotherteamtwice,onceathomeandonceaway,howmanyteamsarethere?
Section9.1Permutations
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PermutationswithRepeatingObjects
Example16
HowmanydifferentwayscanyouarrangethelettersinthewordPOOL?
Example17
HowmanydifferentwayscanyouarrangethelettersinthewordMATHEMATICS?
Insomecases,someoftheitemswewanttoarrangeareidentical.
Ifasetofobjects,withofonekindthatareidentical,ofasecondkind.....theentiresetofobjectscanbearrangedin
Section9.1Permutations
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WithConstraints
twoormoreobjectsmustbeplacedtogether twoormoreobjectscannotbeplacedtogether certainobjectsmustbeplacedincertainpositions
Example18
Howmanywayscanagroupof5peoplebearrangedinalineiftwoofthemaregoodfriendsandwanttosittogether.
Permutations
whencertainitemsaretobekepttogether,treatthejoineditemasiftheywereonlyoneobject.
Mary Beth
Dave Kyle
John
Section9.1Permutations
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Example19
Howmanywayscanagroupof5peoplebearrangedinalineifMaryandBethshouldnotsittogether.
total#ofarrangementswithnorestrictions
arrangementswithMaryandBethtogether
Complement
Section9.1Permutations
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ArrangementsinvolvingCases
Someproblemshavemorethanonecase.Calculatethenumberofarrangementsforeachcaseandthenaddupthevaluesforallcasestoobtainthetotal.
Example20
Determinethenumberofarrangementsof4girlsand3boysinarowofsevenseatsiftheendsoftherowsmustbeeitherbothfemaleorbothmale.
KeyWords:AtLeast,AtMost,Either
CaseforFemales CaseforMales4girls,3boys
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