Chapter1PermutationsandCombinations排列和组合.docx

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Chapter1PermutationsandCombinations排列和组合.docx

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Chapter1PermutationsandCombinations排列和组合.docx

Chapter1PermutationsandCombinations排列和组合

Chapter1PermutationsandCombinations排列和組合

[1.1TheMultiplicationPrincipleofCounting]基本乘法原理

ExampleIntravelingfromHongKongtoLosAngeles,Mr.WongwishestostopoverinHawaii.Ifhehas4airlinestochoosefrominthetripfromHongKongtoHawaiiandhas3airlinestochoosefrominthetripfromHawaiitoLosAngles,inhowmanywayscanMr.WongtravelfromHongKongtoLosAngles?

Solution

ExampleArestaurantoffers2differentsoups,5differentmaincoursesand4differentdrinksinitslunchmenu.Eachordercanselectasoup,amaincourseandadrinkfromthemenu.Howmanydifferentlunchesarepossible?

[菜單：

不同種類的湯、主菜及甜品，可組合成多款套餐]

SolutionSoupMainCourseDrink

∙C∙

A∙∙D∙∙H

∙E∙∙I

B∙∙F∙∙J

∙G∙∙K

TheMultiplicationPrincipleofCounting

Ifafirstoperationcanbeperformedinn1ways,asecondoperationinn2ways,athirdoperationinn3ways,andsoforth,thenthesequenceofkoperationscanbeperformedinn1n2⨯…⨯nkways.

ExampleAladyhas5blouses,6skirts,3handbagsand7pairsofshoes.Howmanydifferentoutfitscanshewear?

SolutionThedifferentnumberofoutfits=

Exercise1A

1.Thereisa6-digitPersonalIdentificationNumber（PIN）encodedineachbankcardforsecurityreasons.FindthenumberofpossiblePINs

（a）withrepeateddigitsallowed,

（b）withnorepeateddigits.

2.InordertotravelfromcityXtocityY,Peterhasthefollowingchoices:

A.HecanflydirectlyfromXtoYononeofthe3airlines.

B.HecanflyfromXtoanothercityTononeofthe4airlinesandthentravelfromTtoYwithoneofthe5buslines.

（a）InhowmanywayscanhetravelfromXtoY?

（b）FindtheprobabilitythatPeterwillgofromXtoYthroughcityT.

3.（a）Inhowmanywayscan5true-falsequestionbeanswered?

（b）Inhowmanywayscan10multiplechoicequestions,eachwith4options,beanswered?

（c）Atestpaperconsistsof5true-falsequestionsand10multiplechoicequestionswith4options.Inhowmanywayscanthispaperbeanswered?

（d）Assumingthatthepassingmarkis50andeachquestioncarries1mark,whatistheprobabilitythatJohncanpassthetestbyjustguessingtheanswers?

4.Monicahasseveralwaystospendherevening.Shecanreadabook,watchavideoorgoforadrink.Shecanchoosebetween7books,5videosand3coffeeshops.Howmanychoicesdoesshehaveif

（a）Sheonlytakesoneactivity?

（b）Sheonlytakestwoactivity?

（c）Shetakesallthreesortsofactivities?

（d）Shetakes1or2orallthreesortsofactivities?

（e）Whatistheprobabilitythatshehaswatchedavideotape?

5.Donna’scosmeticboxconsistsof8coloursofeyeshadow,3lipsticksand4typesofblush.Shemakesupusingeither1,2or3kindsofthesefeatures.Howmanyfacialappearancesarepossible?

6.Athree-digitnumberisformedfromthedigits0,1,3,5,6,7and8.Eachdigitcanbeusedbyonce.

（a）Howmanypossiblenumberscanbeformed?

（b）Howmanyoftheseareevennumbers?

（c）Howmanyofthesearelessthan500?

（d）Whatistheprobabilitythattheformednumberisanevennumber?

（e）Whatistheprobabilitythattheformednumberislessthan500?

7.Avehiclelicenceplateconsistsof2lettersfollowedby4digits.

（a）Howmanydifferentplatesarepossiblewithnorestriction?

（b）HowmanydifferentplatesarepossibleifthelettersOandIareexcluded?

（c）HowmanydifferentplatesarepossibleifthelettersOandIareexcludedandatleastoneofthedigitsis8?

（d）WiththerestrictionofexcludingthelettersOandI,whatistheprobabilitythatthelicencenumberhasatleastoneofthedigitsbeing8?

[1.2Permutations排列]

[[1.2.1PermutationofnDistinctObjects]]

ExampleAboyhas5differenttoysoldiers.Inhowmanywayscanhearrangethemtostandinaline?

SolutionTherequirednumberofways=__________________________=__________

[[[Thefactorialnotation]]]

Now,weintroduceanewnotationn!

readas“nfactorial”.

Forexample,4!

=_______________________=_________

=7⨯6!

Therefore,

Wedefine0!

=1.

Thenumberofpermutationsofndistinctobjectsisn!

Theorem

Example（a）Inhowmanywayscan5studentsbeseatedonarowof5chairs?

（b）Inhowmanywayscan5studentsbeseatedonarowof3chairs?

（c）Inhowmanywayscan5studentsbeseatedonarowof2chairs?

（d）Inhowmanywayscan5studentsbeseatedonarowof1chair?

Solution（a）Thetotalnumberofways=

（b）Thetotalnumberofways=

（c）Thetotalnumberofways=

（d）Thetotalnumberofways=

…

Totalnumberofarrangements=60

[[1.2.2ThePermutationSymbol]]

Ingeneral,thenumberofpermutationsofndistinctobjectstakenratatime,denotedbythesymbol

isgivenby

=n（n–1）（n–2）⋅⋅⋅⋅（n–r+1）

e.g.

=5（5–1）….（5–3+1）=________

=___________________=________

But,

n（n–1）（n–2）⋅⋅⋅（n–r+1）⨯

e.g.

=___________________

ExampleAgirlhas3Chinesebooksand7Englishbooks,allofwhicharedifferent.

（a）6ofthemonashelf?

（b）The3Chinesebooksontheleftandthe7Englishbooksontherightofashelf?

（c）AllofthemonashelfwiththeChinesebookstogether?

Solution

[[1.2.3PermutationofnObjects,notallDistinct]]

ExampleInhowmanywayscanthelettersoftheword“WEEKEND”bearranged?

SolutionWithoutanyfurtherconsideration,wemaythinkthattheansweris________.

However,itiswrong.Why?

Wereplacethe3E’softheword“WEEKEND”withE1,E2andE3.i.e“WE1E2KE3ND”.

wayswillincludethefollowingcases

E1E2E3WKNDE1E3E2WKNDE2E1E3WKNDE2E3E1WKND

E3E1E2WKNDE3E2E1WKND

However,theyarethesameword.Therefore,wemustdeduce7!

by6.Inthesameway,

Therequirednumberofpermutations=

Thenumberofdistinctpermutationsofnobjectsofwhichn1areofonekind,n2ofasecondkind,…,andnkofakthkindis

Theorem

ExampleAshopwindowdesignerhas7balloons,ofwhich1iswhite,2areblueand4arered.Shehangstheseballoonsinalineintheshopfront.Findthenumberofarrangementsshecanmakebyusing

（a）all7balloons,

（b）exactly6balloons.

Exercise1B

1.Evaluate

2.FourplayingcardsClubA,HeartJ,SpadeQandDiamondKarearrangedinarow.

（a）Listallthepossiblepermutations.

（b）Howmanydifferentpermutationsarethere?

3.Amy,Betty,Cathy,DorothyandEmilyarethefinalcontestantinabeautycampaign.

（a）Listthewaysthat3ofthemcanbearrangedinarowtotakephotos.

（b）Inhowmanywayscanthefirstthreeplacesbefilled?

4.Ababyhas4red,1blue,1greenand1yellowcubes.Inhowmanydifferentwayscanthesecubesbestackedup?

5.Howmanydistinctpermutationscanbemadefromthelettersoftheword“EXCELLENT”?

（c）Findthesumofallthenumbersformedin（a）.

11.SusandecoratesherChristmastreeusingaseriesoflightbulbs.Shehas5red,4yellowand2bluebulbsavailable.

（a）（i）Ifsheusesallthelightbulbs,howmanydifferentarrangementscanshemake?

（ii）In（i）,howmanyofthearrangementswillthetwobluebulbsbeseparated?

（b）Ifsheonlyuses10ofthebulbs,howmanydifferentarrangementscanshemake?

[1.3Combinations組合]

Permutationsareconcernedwiththeorderinwhichobjectsarearranged.However,inmanyproblemsweonlyfocusonthenumberofwaysofselectingrobjectsfromgivennobjectswithoutregardtoorder.Theseselectionsarecalledcombinations.

ExampleAdebateteamof3studentsisselectedfromstudentA,B,CandD.Findthenumberofpossiblecombinations.

Solution

Permutations

Combinations

Repetition

ABC,ACB,BAC,BCA,CAB,CBA

Thenumberofpossiblecombinations=

=_____________

[[1.3.1TheCombinationSymbol]]

Thenumberofcombinationsofrobjectschosenfromndistinctobjectsis

Theorem

ExampleFindthevaluesofthefollowing:

（a）

（b）

（c）

[[1.3.2Usefulrelationsforcombinations]]

Frome.g.10,itindicatesthat_______=________.Trytocompute

and

.Whatcanyouseefromtheresults?

Theyindicatesthat_____________=______________and______________=______________.

Why?

Weconsider

===

Hence,wehave

Example（a）Showthat

（b）Usingtheresultin（a）,compute

Solution

ExampleFindthenumberofpossiblecombinationsof49cardsfromadeckof52playingcards.

SolutionTherequirednumberofcombinations=

ExampleThereare10differentbottlesofredwineand12differentbottlesofwhitewineinstock.

（a）Ashopkeeperdeterminestodisplay2bottlesofredand3bottlesofwhitewineonarack.Findthenumberofpossiblearrangementshecanmake.

（b）Nowtheshopkeeperonlywantstodisplayany5bottlesofwine.Whatisthenumberofpossiblearrangementsthathecanmake?

（c）Whatistheprobabilitythatthereare2bottlesofredand3bottlesofwhitewineonarack?

Exercise1C

Evaluate1.

3.Twocoloursarechosenfromthecoloursred,yellow,greenandbluetobethecoloursofalogo.

（a）Listthepossiblecombinationsoftwocolours.

（b）Howmanycombinationsoftwocoloursareavailable?

（c）Whatistheprobabilitythatoneofthechosencoloursisred?

4.AMarkSixlotteryticketconsistsofmarking6differentnumbersrangingfrom1to47.

（a）Howmanydifferentlotteryticketscanyoumark?

（b）Whatistheprobabilitythat

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