Download Section 4-6 EXAMPLES FACTORIALS FACTORIAL RULE

FUNDAMENTAL COUNTING
RULE
Section 4-6
Forasequenceoftwoeventsinwhichthefirst
eventcanoccur waysandthesecondevent
canoccur ways,theeventstogethercan
occuratotalof
ways.
Counting
Thisgeneralizestomorethantwoevents.
EXAMPLES
1. Howmanytwoletter“words”canbeformedifthe
firstletterisoneofthevowelsa,e,i,o,uandthe
secondletterisaconsonant?
2. OVERFIFTYTYPESOFPIZZA!saysthesignasyou
driveup.Insideyoudiscoveronlythechoices
“onions,peppers,mushrooms,sausage,anchovies,
andmeatballs.”Didtheadvertisementlie?
3. Janethasfivedifferentbooksthatshewishesto
arrangeonherdesk.Howmanydifferent
arrangementsarepossible?
4. SupposeJanetonlywantstoarrangethreeofher
fivebooksonherdesk.Howmanywayscanshe
dothat?
FACTORIAL RULE
Acollectionof distinctobjectscanbe
arrangedinorder ! differentways.
FACTORIALS
!
∙
1 ∙
2 ∙ ⋯∙ 3 ∙ 2 ∙ 1
NOTE:0!isdefinedtobe1.Thatis,0!=1
PERMUTATIONS
Apermutation isanordered arrangement.
Apermutationissometimescalledasequence.
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PERMUTATION RULE
(WHEN ITEMS ARE ALL
DIFFERENT)
Thenumberofpermutations(orsequences)of
r itemsselectedfromn availableitems(without
replacement)isdenotedby
andisgivenby
theformula
!
!
EXAMPLE
Suppose8peopleenteraneventinaswim
meet.Assumingtherearenoties,howmany
wayscouldthegold,silver,andbronzeprizes
beawarded?
PERMUATION RULE
CONDITIONS
• Wemusthaveatotalof different items
available.(Thisruledoesnot applyifsome
itemsareidentical toothers.)
• Wemustselect ofthe itemswithout
replacement.
• Wemustconsiderrearrangementsofthe
sameitemstobedifferentsequences.(The
arrangementABC isthedifferent fromthe
arrangementCBA.)
PERMUTATION RULE
(WHEN SOME ITEMS ARE
IDENTICAL TO OTHERS)
Ifthereare itemswith alike, alike,...,
alike,thenumberofpermutationsofall
itemsis
!
!
EXAMPLE
Howmanydifferentwayscanyourearrange
thelettersoftheword“level”?
! ⋯
!
COMBINATIONS
Acombination isaselectionofobjects
withoutregardtoorder.
2
COMBINATIONS RULE
Thenumberofcombinationsofr items
selectedfrom differentitemsisdenotedby
andisgivenbytheformula
!
! !
NOTE:SometimesnCr isdenotedby
.
COMBINATIONS RULE
CONDITIONS
• Wemusthaveatotalof differentitems
available.
• Wemustselect ofthoseitemswithout
replacement.
• Wemustconsiderrearrangementsofthe
sameitemstobethesame.(The
combinationABC isthesameasthe
combinationCBA.)
EXAMPLES
EXAMPLE
1. Fromagroupof30employees,3aretobe
selectedtobeonaspecialcommittee.In
howmanydifferentwayscanthe
employeesbeselected?
2. IfyouplaytheNewYorkregionallottery
wheresixwinningnumbersaredrawn
from1,2,3,...,31,whatistheprobability
thatyouareawinner?
3. Exercise34onpage186.
Supposeyouaredealttwocardsfromawell‐
shuffleddeck.Whatistheprobabilityofbeing
dealtan“ace”anda“heart”?
PERMUTATIONS VERSUS
COMBINATIONS
Whendifferentorderingsofthesameitems
aretobecountedseparately,wehavea
permutation problem,butwhendifferent
orderingsareNOT tobecounted separately,
wehaveacombination problem.
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