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PERMUTATIONS AND

COMBINATIONS

AppliedSta+s+csandCompu+ngLab

IndianSchoolofBusiness

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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