(2) permutations and combinations
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7/30/2019 (2) Permutations and Combinations
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AppliedSta+s+csandCompu+ngLab
PERMUTATIONS AND
COMBINATIONS
AppliedSta+s+csandCompu+ngLab
IndianSchoolofBusiness
7/30/2019 (2) Permutations and Combinations
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LearningGoals
• Basicprincipleofcoun6ng• Combina6ons
• Permuta6ons
2
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AnExample
Amanislookingtomake3investments.Hewishestoinvestinoneofthefollowingtypesofassets:GovernmentBonds,MutualFunds,LandandPreciousMetals.
Hehasnarroweddownto3differentGovernmentBonds,3differentMutualFundsandGold,SilverandPla6numunderPreciousMetals.
Howmanyop6onsdoeshehave?Wecantrycoun6ngthem!
Itisnotanimpossibletaskbutitisgoodtoknowthat
thereisasimpleryeteffec6vemethodtocountthenumberofpossiblecombina6onsoftheseassets
3
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AnExample(Contd…)
InvestmentsCombina6on
GovernmentBonds
BondA BondB BondC
MutualFunds
MutualFundA
MutualFundB
MutualFundc
PreciousMetals
Pla6num Gold Silver
4
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ThePrincipleofCoun6ng
• ThePrinciplestatesthat:If‘p’tasksareperformed,eachwith‘n p’ possible
waystocomplete,thenthereareatotalofwaysinwhichthe‘p’taskscanbecompleted
Usingthisprinciplewehave:
– 3differentassets(Bonds,MutualFundsandMetals)
– 3differenttypesineachasset –
Sowehavea3×3×3=27op6ons! – Sotheinvestorcaninvestin27differentways
5
∏=
p
i
in
1
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Permuta6onAfirmhasresourcesenoughtoinvestin3possibleopportuni6es.Thereare4possible
opportuni6es:A,B,CandDEachoftheseopportuni6esneeddifferentamountsofini6alinvestments.
Oncethecompanyselectsthefirstopportunity,thenextonewilldependonhow
muchresourcesarele\a\erpayingforthefirstone.
ItmustnowchooseANY3outoftheseinsuchamannerthatitbalancesoutits
resources.
Howmanypossibleop6onsdoesthefirmhave?Letuslistafew:
A B C
A C B
B C A
C B A
………… ……… ………
Theseareonlyafew.Theorderin
whichtheseopportuni6esare
selectedisimportantbecausethe
resourceshavetobedistributed
accordingly.Wecalleachofthese
arrangementsaPermuta+on,i.e.
(A,B,C)isonepermuta6on,(C,A,B)isanother.
Now,theques6onishowdowe
countthenumberofpermuta6ons
withoutlis6ngthemall?
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NumberofPermuta6ons/Arrangements
• Letusapplytheprincipleofcoun6ng:
– Thefirstopportunitycanbeanyofthe4
– Thesecondanyoftheremaining3
– Thethirdoneoftheremaining2
– Sowehave4×3×2=24op6ons!
• Ingeneral,thenumberofpermuta6onsof‘n’objectsis
• Supposethatthereare20opportuni6estochoosefrom.Howmanydifferent
permuta6onsof3dowehavenow?
• Usingthesameprincipleasabovewehave:
• Ingeneral,thenumberofpermuta6onsofsize‘r’fromatotalof‘n’objectsis
givenby:
7
!12.).........2()1( nnnn =××−×−×
nPr=
n!
(n− r)!
20×19×18 =20!
17!=
20!
(20 −3)!
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Combina6on
• Supposethatalltheopportuni6eswereequalininvestmentandreturn
amounts,thefirmthenwouldnotdifferen6atetheorderinwhichthe
opportuni6eswereselected
• Inthiscase(A,B,C)isthesameas(C,A,B),i.e.(A,B,C),(A,C,B),(B,A,C),
(B,C,A),(C,A,B)&(C,B,A)areallthesametothefirm
• Theinterestisin{A,B,C}
• ThisiscalledaCombina+on,theorderinwhichitemsareselecteddoes
notmaer,wearesimplyselec6ngaset
• Ouronlyop6onshereare:
8
A B C
A B D
A C D
B C D
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NumberofCombina6ons• Letussayweneedtoselect‘r’itemfrom‘n’
• Thenumberofpermuta6onsis:
• Weknowthatthesealreadyincludetheselec6onandthattheselec6oniscounted
morethanonce
• Aselec6onofsize‘r’hasr!differentpermuta6ons
• Sotoadjustfortheextracoun6ngwedividebyr!
• Therefore,thenumberofpossibleselec6onsis
• Thisisdenotedby
9
n!
(n− r )!
n!
(n− r )!×
1
r !
nC
r or
n
r
!
"#
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Thankyou