Permutations and Combinations

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Career Cafd,CAT " MAT * CSAT * GRE * GMAT

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Permutations and Combinations ' 'i\f '

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7. 12to * 15po =8. 1Bo, a 19c, =g.

, 5o, +5cs =

10. '10c0 + 10., * t}cz + .... + 10.1011. 6!+5! =

10^^12. -

16cz

A ?' 10!lc' e'. , +l

14 11ps3!

.t tr tzl'' v' 4t x4l x4l

16' '71 t *ot5! 2t

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17 . lf n., = 36, tidO nZ, l':'

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18. lf hp, = 990, find n?19. lf n., a ilcz + Ilcs + .... + 11cn = 5'l 1, find n?

2fr. For whirt value of n, n., *'trp, = 570.21. lf n., * nrr= 28, find n ?

4r,* 5., * 6r,* 7r,* Br,=2g. '5.r* 6.r* 7rr* B.r= !.,.24. lt (2n* 1).. = 35, finO ne\tj ,' i '/25. lf h." t hcz + ncs + oc+ = 162,find n? /\ ' \Ll vZ v3

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19. For what value of n, Y = 56?I

20. 'lf h., = h.ro, fl =

21 . 15rn =22. 20o, =23. 2Ics 2lc, =

2ocrz24.

2Oce

. .25. 1l+21 +3! +4! +5! +6! = .:::,.,

Summary ;'1. lf an event 'A' can be done in 'm' ways and an otherr etdht 'B' can be done in

'n' ways, thenexactly one of the events'A', 'B'can be done in 'm + n'ways. (Addition rule)

both the events 'A' and 'B' together can be done in a given order in m x nways. (Multiplication rule)

2. 'n' objects can be ananged in 'n' places in n! ways..n!

3. 'n' objects out of which 'r' are identical, can be arranged in 'n' places in ;ways.

4. 'n' objects can be distributed to 'n' persons such that each gets one in n.!ways.

5. 'n' objects,can be distributed to 'n' persons (such that more than one can begiven to a person) in nn ways.

6. 'r' objects can be arranged in 'n' places (r < n) in npr ways.7. 'r'. objects can be distributed to 'n' persons (r < n) such that no person gets

more than one in npr ways.8. 'r' objects can be distributed to 'n' persons (such that more than one can be

given to a person) in n'ways.9. 'n' persons can stand in a row in n! ways.10.'n' persons can sit around a circular table'in (n

- 1)! ways.

11.'r' objects can be selected out of 'n' objects (i.< n) in ncr ways.

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12.'n' identical objects can be distributed to 'r' persons in (n + r -

l).fr_rlways. \$t i ' r;1,{',3.' 'll^ l.,,,,r,' &;i,', 16-

13. (m + n + p) items, can be divided into three groups of m, n and p items in/m* n * P)._ x (n + P). - (m+n+p)!z/ '"m n m!n!P!

,/,4.(m + n + p) items, can be distributed among three persons such that they getffi,nandp,itemsin (m+n*p)._ x (n+p).,, x 3l = (illl?)'3! -tm!n!p! \

15.The number of ways of selecting one or more objectq out of '6l,r661ects is ,r, l,

(since h.o* Ilcr * frcz+ .... * Ilcn @tl ry\r- i (1., ,'

1 *C l !.'''' V'tt-'- (} ''- 'i16.The maximum number of lines that can be drawn usin! 'n' nbii-collifrSr points

is h., .17 -The maximum number of lings thd:can+e-qfatun.Jrsing '3,go^ints out of whjgh.

* P i{ 0/.? r 1-Qn.' :..q. i.r ':' I,rr.''r'points are collingAf ir 1., *:jg-l -1-',,,,t,,,,,, , , I18.The maximum numb"t offi-i"ngles tfiai canrpe,formed using'n'non collinear

points is I1., . '

t g. fne maximum number of triangles that can be formed using 'n' points out of't't- '-2cAL,,t"..c. o-"' >I'r rl.t "..it | .',,....t' 'r '''which 'r' pqints are collinear is lla,

-

I.cs !" r'Y..L'v'\ ?" -'20. The maximum number of inteisectioh points formed by drawing 'n' lines is

ffrr.21.The maximum number of intersection points formed by drawing 'n' circles is

2 x n^ . D'S.fi"l, '""'' l', 'i #u{,. J '1..'p /.' :: ";"t.r/ * r",' r'

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22. 'n' lines, no two of which are parallel and no three of them pass through thesame point are drawn on a plane, Then the number of regions that the planewould be divided into ir (X n) + 1 *

23. The number of diagonals that can be drawn in a polygon of 'n' sides isn(n-3)

rl.^ rr =vz2 uM;gv.+Concept Practice

There are 10 boys and 12 girls participating in a chess tournament. lf each playerplays exactly one game with each.ol:lhe o.theeFlayer,-how ma!]-{f}me\$ate there inwhic rt?

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While packing for a business trip Mr.,Rahul has packed 2 pairs of shoes, 5 pants and6.shirts. The outfit is defined as consisting a pair of shoes, a pant and a shirt. Howmany different outfits are possible? ?Y( y.,"'

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1) 60 2) 40 3)'30 4) 20 5) 13from A to B and six different routes from B to C andD. In how many different ways one can travel from A

2) 30 3) 40 4) 48be distributed to 4 children such that

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1) 20 2) 24 3) 120 4) 625 5) 1024

There are five different routeseight different routes from C toto D via B and C? /

^ '-r tL5) 240 t^ :1) 1eIn how many ways 5 different chocolates canany child can get any number of chocolates?

u 7.

In how many ways 8 different tasks can be assigned to Q,mgn such that each getsongtask? '-r- 'a-1-ogt"y^.''^!6;" '!tI'''"c oLu''" ,''f"' l''-"''1)B 2)16 3)64 4)B! u 5)8tThere are 8 tasks and 8 persons. Task 1 cannot be assigned to person 1.lf everyperson has to be assigned one task, in how many ways this assignment can bedone?1)71 2)7x7t 3)7xB! 4)8! 5)BxB!Five tasks are to be assigned to five persons such that each gets one task. Task 1can not be assigned eitheqto person 1 or to person 2. ln how many ways can theassignment be done? i' . "5: r' 3'.'r t i' ' : '' '' {

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1) 24 2) 48 3) 72 4) e6 5) 120?qY GxL r oy 1

Five Brizes are to be given to five persons such that each gets one. The first prizecan not be given to A or B. The second prize must be given to C. ln how many waysthe distribution of prizes can be done? ' -, ' :. :' "' ''. '' :1) 120

.2) 72 ,., 3) 18 4) 12 5) None of these\ irlr\ i.'' 'There are 15 boys and 12 girls in a class. lf each boy sings a two - minute song oneafter an other with each girl, what is the total duration of the singing?1) 54 minutes 2) 3 hours 3) 4 hours 4) 6 hours 5) None-

Ar

In how many ways six boys can sit in six chairs which are in a row such thatparticu|arboy@:[a!the.extremes?..,.....-.Y..-.(9ii,1) 120 2) 240 ) 360 - 4) 480 5) 720t':1lY LrVr,ll".3 A 7,r \1--z,i lL iL

11.: Eight boys have to be seated in eignt chairs.numbered 1 to 8 in a row. In how manyways this seating can be done such that if a particular boy does not want to sit in thefirsi four chairs and an other boy wants to sit either in 7th or in Bth chair? z '' ' , 1 , .1)6! 2) 6x6! 3) 8x6! 4)6x7 5).7x6!

12' Eight boys have tb Ue seated in eight chairs numbered 1 to B in a row. ln how manyways this seating can be done such that if a particular boy does not want to sit in theeven numbered chair and another boy wants to sit at the extremes?1)gx6t '2)7x6 3)6x6i 4) 5x6! 5)Noneofthese, , ,

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tZ/ Five students A, B, C, D, E have to be given five. ranks .1, 2, ?, 4,,5. A has to be given\'/ either first or second rank. B should be given either: third or fourth rank. C cannot be

given fifth rank. In how many ways this ranking can be done?'

Jx JY iY & I iA test has three sections with 2, 3, 4 questions respectively. One question is to beanSWeredfiomeachsection.|nhowm?l,ydifferentwFySjstudentcan@tn"qUeStiOnS? l-Y L) "f t,Q'{ .ht,"^19, l'f.. J t:'-" .TI1?-T"''.'

r{ 4H* * r'-,21t;tr-,y 2,,-r t =:o Jx,rx s l) 2', n*r :U:ffi-1> How many four digit numbers can be formed usingf the digits 2, 4, 5, 8, if the,-/ repetition of the digits is allowed?

1) 12. 2) 16 3) 24 4) 32 5) 48

1) 16 2) 24 3) 256 4) 40 5) None, of these1Q/ How many four digit numbers can be formed using the digits 1, 4, 8, 9 exactly once?\-/ 1) 4 2) 16 3) 24 4) 256 5) None of these

,at

How many fpur digit numbers can be formed .using the digits 0, 1, 2, 5, 6, if the

repetition of ihe digits is allowed?

' 2)96 5) 625How many four digit even numbers can be formed using the digits 0,

l1v

1, 2, 4, 5, if.therepetition of the digits is allowed?

1) eoo 2) 1080 3) 2160 4) 3240 5) 3888E;-r'.V* \x:1, EY d )/_,!lr ,i,'r :.

.-1,22/H.ow manyfive digit odd numbers can befoimed usihg th6 digitr Li., ?,9,4,O if theV repetition of the diglt^t is ry\$[!ryed?

- -.

tx h f lr f if1) 144 2) 288 3) 216 4) 480 5) None of these ",-\ h xt4xqxJ,r. :/3'?g) . All t'nb toufl6idt numbers that cin be formed using the digits 1, ?,4,.5 lvith 9yl

,./ repg\$g1 are written. How_many olthefn.are divisible by 4? l\ (r ,) n.\.{ DV_-. 2)6 f 4)12 5)18 s;^i, (fr"'rl 'l',

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