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'DC '7I " 'X '7I # 'DE 0_7I $ 0E ' % _ 'EA ' & '_EB_ ' ' '_EC_ ' ( 'ED ' ) 'X( *   (Example:Howmanywayscantheclubelectapresidentandatreasurerif + thepresidentmustbefemale?(Andy,Bill,Cathy,David,Evelyn) , Usethesameproducttable.OmitallresultsthatdonotstartwithCorE.  . (SoomitrowsA,B,D.)Answer:8 6H/ Treediagramsareanotherwaytosystematicallylistthepossibleoutcomes 1 Example:Howmanychoicesofabeverage,asandwich,anddesertdoes 3 Carmellahaveforlunch?Thebeveragechoicesaremilkandjuice.The )4 sandwichchoicesarechicken,tuna,orvegetable.Thedesertchoicesarea D V5 fruitcuporpudding.Answer:12 q!6 wC3/b~0) `  ` @E#` ` Q" w QQ(#(# QQQQ QQQQ QQQQ QQQQ R'd$;QQ @(3  __THEFUNDAMENTALCOUNTINGPRINCIPLE HZ The FundamentalCountingPrinciple providesashortcuttocalculate how  many.  Whenataskconsistsofkseparateparts,ifthefirstpartcanbedoneinn1 ) ; ways,thesecondpartcanbedoneinn2ways,andsoonthroughthe_kth_part, V h  whichcanbedonein_nk_ways,thenthetotalnumberofpossibleresultsfor   completingthetaskisgivenbytheproduct   @ii*ie1!b `@E  i    RevisitthisExample:Howmanywayscanapresidentandasecretarybe  chosenfora5memberclub?(Assumethesamepersoncantbeboth.)  0  0` (#(#5waystochoosethepresident.@R` (#` (#    ` 4waystochoosethesecretary.0 p 0p(#p(#m(#(#    ` 54=20(mucheasier!)  Example:0 ` Howmanynonrepeating3digitnumberscanbeformedfrom  theset{1,2,3}?!3` (#` (# 0  0` (#(#Solution:0` (#` (#Thefirstdigitcanbechosen3ways.{(#(#    `    Theseconddigitcanbechosen2ways.      `    Thethirddigitcanbechosen1way. ! 0  0` (#(#0 ` (#` (#0 (# (#321=6# (#(# Example:0 ` Howmanyeven4digitnumberscanbeformedfromtheset\%n"` (#` (# 0  0` (#(#{0,1,2,3,4}?(Numberscanrepeat.Letsnotallowtheleading &# digittobe0.)'$` (#` (# 0  0` (#(#Thereare4waystochoosethefirstdigit.*"'!` (#` (#    ` Thereare5waystochoosetheseconddigit. =+O("    ` Thereare5waystochoosethethirddigit. j,|)#    ` Thereare3waystochoosethefourthdigit(mustbeeven) -*$    ` 4553=300  .+% ~&X%~ & %%&X@  FundamentalCountingPrincipleWorksheet   1.0  HowmanydifferentoutfitscanEllenwearifshehasthefollowingchoices?(#(#   (a)0 ` 4dresses,3hats,and2pairsofshoesn n B(#` (#` (#(Ans:24) AS   (b)0 ` 5dresses,3hats,and2pairsofshoesn n B(#` (#` (#(Ans:30)     _(c_)0 ` 6dresses,4hats,and5pairsofshoes  A(#` (#` (#(Ans:120)   2.0  Arestaurantofferstwochoicesofsoup(clamchowderorvegetable),fivechoicesofentree(beef, 1C  chicken,ham,liver,orfish),threechoicesofvegetable(potato,corn,orpeas),andthreechoices  ofdessert(icecream,pudding,orsherbet).Ifyoumustselectexactlyoneitemineachcategory,  howmanydifferentcompletemealscouldyouorderinthefollowingsituations?(#(#   (a)0 ` Youcannothavesherbet.n n B(#` (#` (#(Ans:60) Ug   (b)0 ` Youwillnoteatliverorfishastheentree.n n B(#` (#` (#(Ans:54)    _(c_)0 ` Youmustavoidbothclamchowderandliverinthesamemeal.n n B(#` (#` (#(Ans:81)    0 ` Hint:Findthenumberofmealsyoucangetwhenyouorderclamchowder.` (#` (# 0  0` (#(#Thenfindthenumberofmealsyoucangetwhenyouordervegetablesoup.Addthese  numberstogether.q ` (#` (# 3.0  Dan,Alicia,Wendy,Drew,Jennifer,Ben,Gary,andHeatheraretobeseatedinarowofeight i"{& chairs.Inhowmanywayscantheyarrangethemselvesinthefollowingsituations?=#O '(#(#   (a)0 ` Therearenorestrictions.ZZ?(#` (#` (#(Ans:40320) $!) 0  (b)0` (#(#WendyandDrewwillbenexttoeachother.ZZ?(#` (#` (#(Ans:10080) 5(G%- 0  0` (#(#(Hint:FindthenumberofwaysthatWendyandDrewcanbeseatednexttoeachother  )&. bycountingthenumberofwaysthatWendycanbeseatedandthenumberofwaysthat )&/ Drewcanbeseated.Thencountthenumberofwaysthattheother6canbeseated.*'0` (#` (#   _(c_)0 ` Themalesandfemalesaretoalternate,witheitheramaleorafemaleinthefirst.+4` (#` (# 0  0` (#(#chair.@(#` (#` (#(Ans:1152)#&X%%& '#Ԉ  .+5 &N%%&X%&N@(5  FACTORIALS HZ eThenotationn!iscalled nfactorial ,andn!=n(n󀄀1)(n󀄀2)...321  eBydefinition,0!=1.   Examples V h  5!=54321   6!=654321=654=120    3!0  3217I (#(# 6!=6543!=654=120  3!0  3!(#(# Tofind5!OntheTI73:Hit5,thenhittheMATHbutton,scrolloverto * _PRB_,andscrolldowntothe!.ThenhitENTER. EW PERMUTATIONS   Permutation isjustanotherwordforanorderedarrangementoranordered   sequenceofdistinctobjects(repetitionisnotallowed).Youcanthinkofthe &8 word permutationasan arrangement. S e Definition:0 ` Anyorderedsequenceofrobjectstakenfromasetofndistinct " objectsiscalledapermutation.# ` (#` (# Theorem:0 ` Thenumberofwaysofarrangingndistinctobjectstakenrata 4&F# timeisgivenby:a's$` (#` (# @((*ie1!b `@E((i߈ (%  Theformulainthistheoremcanbemadeintuitive.Wehavenobjectstofill T,f)" ther slotsoftheorderedarrangement.ApplytheFundamentalCounting -*# Principletoseehowmanywaysthiscanbedone.  .+$ @(6  Actually,thereisnoneedtomemorizetheformulainthetheorem,because HZ theFundamentalCountingPrinciplecanbeappliedeverytime! u Example0 ` Acontractorbuildshomesof8differentmodelsandhas5lots  tobuildon.Inhowmanydifferentwayscanheplacethemodel   homesontheselots?Thatis,wewishtocountthenumberof ) ; arrangementsof8homestaken5atatime.V h ` (#` (# 0  Solution:0 (#(#    0h (# (#  0h(#h(# p 0(#(#  0x(#(#  x(#x(# 0  0` (#(#0 ` (#` (#Thereare8waystoassignahometothefirstlot.   (# (#   0 ` 0 ` (#` (#Thereare7waystoassignahometothesecondlot.7I  (# (#   0 ` 0 ` (#` (#Thereare6waystoassignahometothethirdlot.dv (# (#   0 ` 0 ` (#` (#Thereare5waystoassignahometothefourthlot. (# (#   0 ` 0 ` (#` (#Thereare4waystoassignahometothefifthlot. (# (#   0 ` 0 ` (#` (#87654=8!=6720* (# (#   0 ` 0 ` (#` (#0 (# (#0h(#(#3!EWh(#h(#   0 ` 0 ` (#` (#(Sameastheformulainthetheorem.)r (# (# @*Example0 ` Alibraryhas9copiesofacertaintextbook(numbered#19),of  whichcopies1,2,3,4arefirstprintings,andcopies5,6,7,8,   and9aresecondprintings.&8` (#` (# 0  0` (#(#(a)0 ` (#` (#Inhowmanywayscanthenumberedtextbooksbe ! arrangedontheshelf?(Ans:9!=362,880)" (# (#   0 ` (b)0 ` (#` (#Inhowmanywayscanthenumberedtextbooksbe (%  arrangedontheshelfifallthefirstprintingsmustbeto )&! theleftofthesecondprintings?(Ans:4!5!=2,880)*'" (# (#   o.+% @(7    0 ` _(c_)0 ` (#` (#Supposethenumberedtextbooksarearrangedonthe HZ shelfatrandom.Whatistheprobabilitythatallthefirst u printingsaretotheleftofthesecondprintings? (# (# 0  0` (#(#0 ` (#` (#Solution:  (# (# 0  0` (#(#0 ` (#` (#Allthearrangementsareequallylikely,andthereare ) ; 362,880ofthem.Thenumberoffavorableoutcomesis V h  2,880.Sotheprobabilityis2880/362880=.0079=.79%  (# (# COMBINATIONS   Thewaysofselectingasubsetofaset,wheretheorderinwhichthe 7I  elementsareselectedisofnoimportance,arecalled combinations . dv (Repetitionisstillnotallowed.)  Thenumberofcombinations(orsubsets)ofndistinctthingstakenrata  timeisgivenby * @ee*ie1!b `@Erri߈ r Example0 ` Youneedtosenda3persongroupfromaclassof5studentsto 8J representtheclassatacontest.Howmanydifferentgroupsare ew there? ` (#` (# 0  0` (#(#ie!"1!b `@E` "` "i߀=10"` (#` (# Example0 ` Howmany5cardhandscanbedealtfroma52carddeckof &# cards?'$` (#` (# 0  0` (#(#ie 1!b `@E` 9*` 9*i߀=2,598,9609*K'` (#` (#   ,/>," @(8  0  0` (#(#Howmany5cardhandsofalldiamondscanbedealtfroma52 HZ carddeck?u` (#` (# 0  0` (#(#ie1!b `@E` ``` `i߀=1287` (#` (# 0  0` (#(#Whatistheprobabilityofbeingdealtahandofalldiamonds? ` (#` (# OntheTI73calculator,youcangetie#1!b `@E%%i߀withafewkeystrokes.  0  Hit13,thenhittheMATHbutton,andscrolloverto_PRB_.Scrolldown  to_nCr_ԀandhitENTER.Thenhit5andENTER.(#(#    o& %o @CountingTechniquesWorksheet   Amarriedcouplekeeps4actionmoviesand5romanticcomediesbytheDVDplayer.  (a)0  Inhowmanywayscanthemoviesbearrangedinarow? C(#(#(#Ans:9! AS (b)0  Inhowmanywayscanthemoviesbearrangedinarowif:9 K (#(#   (1)0 ` alltheactionmoviesmustbetotheleftoftheromanticcomedies?>(#` (#` (#Ans:4!5!     (2)0 ` thefirstandlastmoviesmustberomanticcomedies?  ;(#` (#` (#Ans:547!  _(c_)0  Howmanywayscanthecouplechoose2moviestowatchthatnight? C(#(#(#Ans:36 y (d)0  Howmanywayscanthecouplechoose2moviestolendtotheirnextdoorneighborandanother EW! 2moviestolendtotheneighboracrossthestreet?0 0 A(#(#(#Ans:756 +" 0  Hint:Thisisatwoparttask.Ifthefirstpartcanbedoneinn1waysandthesecondpartcanbe  $ doneinn2ways,thenthetotalnumberofpossibleresultsforcompletingthetaskisn1n2(bythe !% FundamentalCountingPrinciple).i"{&(#(# (e)0  Ifthehusbandisforcedtochoose3romanticcomediestobewatchedthatweekend,howmany 5(G%- differentchoicesdoeshehave? B(#(#(#Ans:10  )&.