cbse class 11 maths notes _ permutations and combinations

10/27/2014 CBSE Class 11 Maths Notes : Permutations and Combinations

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CBSE Class 11 Maths Notes :Permutations and Combinations

May1,2014byDileepSingh

FundamentalPrinciplesofCounting

1.MultiplicationPrinciple

Iffirstoperationcanbeperformedinmwaysandthenasecondoperationcanbeperformedinnways.Then,thetwooperationstakentogethercanbeperformedinmnways.Thiscanbeextendedtoanyfinitenumberofoperations.

2.AdditionPrinciple

Iffirstoperationcanbeperformedinmwaysandanotheroperation,whichisindependentofthefirst,canbeperformedinnways.Then,eitherofthetwooperationscanbeperformedinm+nways.Thiscanbeextendedtoanyfinitenumberofexclusiveevents.

Factorial



Foranynaturalnumbern,wedefinefactorialasn!orn=n(n1)(n2)3x2x1and0!=1!=1

Permutation

Eachofthedifferentarrangementwhichcanbemadebytakingsomeorallofanumberofthingsiscalledapermutation.

MathematicallyThenumberofwaysofarrangingndistinctobjectsinarowtakingr(0rn)atatimeisdenotedbyP(n,r)or p

PropertiesofPermutation

ImportantResultsonPermutation

1. Thenumberofpermutationsofndifferentthingstakenratatime,allowingrepetitionsisn .2. Thenumberofpermutationsofndifferentthingstakenallatatimeis P =n!.3. Thenumberofpermutationsofnthingstakenallatatime,inwhichparealikeofonekind,q

arealikeofsecondkindandrarealikeofthirdkindandrestaredifferentisn!/(p!q!r!)4. Thenumberofpermutationsofnthingsofwhichp arealikeofonekindp arealikeofsecond

kind,p arealikeofthirdkind,,P arealikeofrthkindsuchthatp +p +p ++p =nisn!/P !P !P !.P !

5. Numberofpermutationsofndifferentthingstakenratatime,

nr

r

nn

1 2

3 r 1 2 3 r

1 2 3 r

n1



whenaparticularthingistobeincludedineacharrangementisr. P .whenaparticularthingisalwaysexcluded,thennumberofarrangements= P

6. Numberofpermutationsofndifferentthingstakenallatatime,whenmspecifiedthingsalwayscometogetherism!(nm+1)!.

7. Numberofpermutationsofndifferentthingstakenallatatime,whenmspecifiedthingsnevercometogetherisn!m!x(nm+1)!.

DivisionintoGroups

(i)Thenumberofwaysinwhich(m+n)differentthingscanbedividedintotwogroupswhichcontainmandnthingsrespectively[(m+n)!/m!n!].

Thiscanbeextendedto(m+n+p)differentthingsdividedintothreegroupsofm,n,pthingsrespectively[(m+n+p)!/m!n!p!].

(ii)Thenumberofwaysofdividing2ndifferentelementsintotwogroupsofnobjectseachis[(2n)!/(n!) ],whenthedistinctioncanbemadebetweenthegroups,i.e.,iftheorderofgroupisimportant.Thiscanbeextendedto3ndifferentelementsinto3groupsis[(3n)!/((n!) ].

(iii)Thenumberofwaysofdividing2ndifferentelementsintotwogroupsofnobjectwhennodistinctioncanbemadebetweenthegroupsi.e.,orderofthegroupisnotimportantis

[(2n)!/2!(n!) ].

Thiscanbeextendedto3ndifferentelementsinto3groupsis

[(3n)!/3!(n!) ].

Thenumberofwaysinwhichmndifferentthingscanbedividedequallyitintomgroups,iforderofthegroupisnotimportantis

[(mn)!/(n!) m!].

(v)Iftheorderofthegroupisimportant,thennumberofwaysofdividingmndifferentthingsequallyintomdistinctgroupsismn

n1r1

n1r

2

3

2

3

m

m



[(mn)!/(n!) ]

(vi)Thenumberofwaysofdividingndifferentthingsintorgroupsis

[r C (r1) + C (r2) C (r-3) +...].

(vii)Thenumberofwaysofdividingndifferentthingsintorgroupstakingintoaccounttheorderofthegroupsandalsotheorderofthingsineachgroupis

P =r(r+l)(r+2)(r+n1).

(viii)Thenumberofwaysofdividingnidenticalthingsamongrpersonssuchthateachgets1,2,3,orkthingsisthecoefficientofx intheexpansionof(1+x+x ++X ) .

CircularPermutation

Inacircularpermutation,firstlywefixthepositionofoneoftheobjectsandthenarrangetheotherobjectsinallpossibleways.

(i)Numberofcircularpermutationsatatimeis(n-1)!.Ifclockwisetakenasdifferent.ofnanddifferentthingstakenanti-clockwiseordersallare(ii)Numberofcircularpermutationsofndifferentthingstakenallatatime,whenclockwiseoranti-clockwiseorderisnotdifferent1/2(n1)!.(iii)Numberofcircularpermutationsofndifferentthingstakenratatime,whenclockwiseoranti-clockwiseordersaretakeasdifferentis

P /r.

(iv)Numberofcircularpermutationsofndifferentthingstakenratatime,whenclockwiseoranti-clockwiseordersarenotdifferentis

P /2r.

(v)Ifwemarknumbers1tononchairsinaroundtable,thennpersonssittingaroundtableisn!.

m

n r1

n r2

n r3

n

n+r-1n

nr 2 k-1 r

nr

nr



Combination

Eachofthedifferentgroupsorselectionswhichcanbemadebysomeorallofanumberofgiventhingswithoutreferencetotheorderofthethingsineachgroupiscalledacombination.

MathematicallyThenumberofcombinationsofndifferentthingstakenratatimeis

PropertiesofCombination

ImportantResultsonCombination

Thenumberofcombinationsofndifferentthingstakenratatimeallowingrepetitionsisn+r

1



CThenumberofwaysofdividingnidenticalthingsamongrpersonssuchthateachonegetsatleastoneis C .Thetotalnumberofcombinationsofndifferentobjectstakenratatimeinwhich(a)mparticularobjectsareexcluded= C(b)mparticularobjectsareincluded= CThetotalnumberofwaysofdividingnidenticalitemsamongrpersons,eachoneofwhomcanreceive0,1,2ormoreitems(n)is CThenumberofwaysinwhichnidenticalitemscanbedividedintorgroupssothatnogroupcontainslessthaninitemsandmorethank(mP5. Thenumberofwaysinwhichnidenticalthingscanbedistributedintordifferentgroupsis

C ,or C accordingasblanksgroupsareorarenotadmissible.6. Thenumberofwaysofansweringoneormoreofnquestionsis2 1.7. Thenumberofwaysofansweringoneormorenquestionswheneachquestionhasan

alternative=28. n!+1isnotdivisiblebyanynaturalnumberbetween2andn.9. Ifthereare1objectsofonekind,mobjectsofsecondkind,nobjectsofthirdkindandsoon.

Then,thenumberofpossiblearrangementsofrobjectsoutoftheseobjects=Coefficientofx intheexpansionof

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cbse class 11 maths notes _ permutations and combinations

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