Permutations and Combinations

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1 x2 x 3 x.... x n

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n!

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harr-,

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2.

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.4.

5.,

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1Oce

tZct

15cr +

8.,

50,

Speed -1

=

L5c2 + 15.. =

6r, * 6o,

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Help line No. 95012€50{2Page 1 of 26

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

7. 12to * 15po =

8. 1Bo, a 19c, =

g. , 5o, +5cs =

10. '10c0 + 10., * t}cz + .... + 10.10

11. 6!+5! =

10^^12. -16cz

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

14 11ps

3!

.t tr tzl'' v'

4t x4l x4l

16' '71 t *ot5! 2t

t

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 .v+

d,.r,

Career Cafd,The simplest way to crack CAT

Help line No. 95012-55012Page 2 of 26

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Ca re,er Cafd,CAT * MAT * CSAT * GRE * GMAT

1.8cs

8cz

lfk=4,findn?["t

ff (n * L) r, ' rrcz

tZcz Lzcs

lf Znrn =20, find

ff Bc. =70, find r?

5! x5! "

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4pi , :': :: ':"

n".,r. : ''::

LZps

12cs

1,3c2 + L4c2 +

t3p2 + L4p2 +

lf lpr = 720, findtr.,

6lx7o'.

6!x3! =

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=

=5:4,thenn=

n?

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12:.

13.

ff n.r, * [pr, = 721,find

n.o * trpo

lf n.o * [.r,,i

Find n.I,"'if" nt'

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Ca reer Cafd,CAT * MAT * CSAT * GRE * GMAT

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 n

ways. (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 be

given 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 begiven 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.

Career Cafd,The simplest way to crack CAT

Help f ine No. 95012-55012Page 4 of 26

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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 get

ffi,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,goînts out of whjgh.*

P i{ 0/.? r 1-Qn.' :..q. i.r ':' I,rr.''r'points are collingAf ir 1., *:jg-l -1-',,,,t,,,,,, , , I

18.The maximum numb"t offi-i"ngles tfiai canrpe,formed using'n'non collinearpoints 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 isffrr.

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'

w2t,

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 plane

would be divided into ir (X n) + 1 *23. The number of diagonals that can be drawn in a polygon of 'n' sides is

n(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? <A_^J )1) 10 2) 12 3) 22 4) 120 5) N6nr*- +

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Caree,r Cafd,The simplest way to crack CAT


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While packing for a business trip Mr.,Rahul has packed 2 pairs of shoes, 5 pants and

6.shirts. The outfit is defined as consisting a pair of shoes, a pant and a shirt. How

many different outfits are possible? ?Y( y.,"'- \ ) /J

1) 60 2) 40 3)'30 4) 20 5) 13

from 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) 48

be distributed to 4 children such that*)un^^. '.1 a., - / l'' ' '

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) 1e

In 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 gets

ongtask? '-r- 'a-1-ogt"y^.''^!6;" '!tI'''"c oLu''" ,''f"' l''-"''

1)B 2)16 3)64 4)B! u 5)8t

There 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 1

can 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' ' : '' '' {

3,

v

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 prize

can 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 one

after 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".

3A 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, , , ,t

-y x' ) '''. i 'i -'

Care.e.r Cafd,The simplest way to crack CAT

Hef p line No. 95012-55012Page 6 of 26

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

Jx JY iY & I i

A 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"''.'

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=: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 these

1Q/ 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 therepetition of ihe digits is allowed?

' 2)96 5) 625

How 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 if

1) 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;î, (fr"'rl 'l',t'.... ' ' pv?., X.l ,, bl- " -----'/,,8/ How many four digit numbers which are divisible by 4 can be iormed llsing the digitstV 2, 3, 4, 6 tf the repetition of the digits is atlowe d? ,

l j:

lr

-:- z, J, ct, 9 tllu lEPgl,ll,lull ul Llltt ul9ll,D ID cllluwtiur /. ])u ,6\ ,^ 2)'!,a 3) 6i ^)6 W?6 , ? ,r !, ! .,,jrp(o:tr".r',, '' Vi n, ' 5\ 32 ' 36 /

*W l'y paV riu:.9isrt::i?:i: y.T:l::"-r,:r1?!::iJ '?l,9""rggRd-"ins the disits

0J X 4 EJ 6 if the repetition of the digits is allowed? 3XB X zyq e Fr)Tzoo :..2)1800 3)2J6;9._., 1)2490, ..-5)3600' ?\6v6Xlc"*:{t,l'- 1)1200.'){}2)1800 t)'f''9t.rll 3t31?9-!,:y5)3600 ?.r6Y6)

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_){1) 24 2) 96 3) 120 4) 5p0 .:!,.5) 625 /'| D' r \

,/ h X Sl hls" l,: \- t,, '1'$rr;>

_1{ How many four digit numbers can be form'edi using the digits 0,1 , 2, 4,8, if the\'/ repetition of the digits is not allowed? : ,

3) 120 4) s00ttx \x'3X L

1\72 2) 300 3) 375 - 4) 2q0 5) 250

,9 il:y"fiHJ;::rrisit even ",'o"Jl"" ;: #{r using the digits o,, { {, u,n1 v,, y i, r

v 1) 54 2) 60 3)72 4) 180-,u,u'', " 5) 300 trs

,Ao*many rive disit odd numbers:'*lJr"*J.-.].,*1n31'!,l"to,f,'r, ,,'j, u, n,n"\,/ repetition of the digits is allowed?

/1) 12e6 2) 432 3) 128 4) 120 5) 48

./3V How many four letter words which start with a vowel can be\'r' alphabets A C, D, E, G, H, if the repetition of.the letters is allowed?

Career Cafd\riCAT * MAT * CSAT * GRE * GMAT

.\How many five digit numbers which are multiples of 4 can be formed using the digits0,.1, 2., 4,5, 6 if the repetition of the digits is not-allowed?1.) 180 2) 204 3) 216 {ZqO 5) 320

o,, ffi* ;,-'.,Ifu* \r.'9fran be r"mlXn'fif;" disits 0

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26.l!

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!;Y)27.

30.

,l 31.

1) 12

with out repetition, how many are divisible by 3?1) 24 2) 72 3) e6

2) 14 3) 24 4) 28

4) 1e2 5) 288

5) 32

2, 4,5, 8il

I .a ' r'

3o 4, 6, B without

x 23 x 11111

formed using the

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formed using the

1,

1a

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29.

Ho*^ many slr gjgt.numbers can be formed using some or all of the digits 0, 3, 4, 5?1) 4u -2) 3a 3) 3 x 45 4) 12 x 35 5) None

3 "85.How many four digit numbers which are divisible by 6 can be formed usin!; if,-" Oigitt0, 1, 2, 6,7 with out repeating any of them?

ffiil,?:!:#'06: g:::t;r tnanfl?T.l"" than a mirion can be form:;-l l',':,1:",

1) 216 2) 72e , i: 3) 10.??. 4) 2160 5) 2400 : ;

What is the sum of alt tne foiri ilgit numbers that can be formed using the digits 2, 3,4, 5 with out repetition? .

1) 15554 2) 93324 3) 186648 ,,. 4) 995,456 5) None

What is the surn of all the four digit numbers that can be formed using the digit s 1, 2,3, 4,5 if the repetition of the digits is allowed?1 ) 39e960 2) 999900 3) 1 38875 4) 2083125 5) None

All the five'digit numbers that can be formed using the digits 2,repeating any one of them are written. What is the sum of them?1) 24 x 23 x 11111 2) 256 x 23 x 11111 3) 6254) 120 x 23 x 11111 5) None of the above

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How many four letter words which start with a vowel can bealphabets A, C, D, E, G, H without repetition?

1) 12e6 2) 432 3) 128 - 4) 120

How many four - letter computer passwords can be formed using the alphabets?1) 26. 2) 4" 3) 264 4)26x25x24x23 5)None

How many four - letter computer passwords can be formed using the alphabets if therepetition of the alphabets is not allowed?

4)26x25x24x23 5)None1) 1) 26 '2) 4za q 264

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/38/ l-low many three - letter computer passwords can be formed using the alphabets and\-/ the digits such that the first letter cannot be a digit?

1 ) 33696

t'\ t.

2) 46656

.-, ' ] d

3) 17576 4) 31 850 5) 12960

Care,e.r Cafd,The simplest way to crack CAT

Help f ine No. 95012-55012Page B of 26

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A new flag is to be designed with five vertical stripes using some or all of the ccloqrsOrange, Green and Yellow such that the extreme stripes cannot be green. Then thenumber of ways this can be done is:1) 243 2) 108 3) 81 4) 27 5) 125

A new flag is to be designed with five vertical stripes using some or all of ti're coloursOiange, Green and Yellow such that the adjacent stripes cannot be of the samecolour. Then the number of ways this can be done is:1) 243 2) 108 3) 81 4) 48 5) 12s

In how many ways 4 boys can be selected from 6 boys?1) 6 2) 10 3)15 4) 24 6. 5) 360 '!:,,

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In how many ways 4 students can be selected from 7 studentS such that a particularstudent named Parul must be selected? lr , : li .'- ' . l,i1)35 2)20 3) 18 4)15 ', 5)Q

In how many ways 5 students can be selected from B students such that a particulaistudent named Sumit cannot be selected? rc,1) 70 .2) 56 3) 35 4) 21 5) None \' ')1'". , . -l

There are eight persons. Five persons shouid be selectedpersons should be selected from the first three persons. Inselection can be done? |

1 ) 1 o 2) 20 3) 30 4) 56 5) None of these

45. There are eight persons. Five persons have to be selected such thattEt@TWO)persons have to be setected from the first three persons. tn how many ways #. selection can be done?1) 20 2) 30 3) 40 4) 56 5) None of these

.46. From eight persons A, Bl C, D, E, F, G, H, foui should be selected such that if A isselected, B also should be selected. ln how manyways this can be done? !.;1)50 2)60 ) 3)15 4)30 s)ss +

t '' j

"/ Jtr 'r

.17. From eight personb A: B, C, D, E, F, G, H, four should be setected such that if A is\.qe|ected,Ccannotbese|ected.HowmanyWaySthiscanbedone?4\ An '.)\ An o\ ,lE ,\ "tn F\ FF l' rr ' ll ,r'., ''t -- a l 1Ft -I vv t, I w,v,

$ 'r'' titr'l

9' ln how many ways four persoris can be selected out of.eight persons A, B, C, D, f,F,.G,Hsuchthatifane!-eurvnCisse|ected,Dhasto-bese|ected?.'1)50 2)60 3)15 4)50- ') ,',, .- . , {,,. _

49-r' - A team of five is to be selected from nirie persons {, B, C, D, E, F, G, H, I such thatthe team must include e.[h4pr-Eljut not both?1) 35 2) 37 -g) OO 4) 70 (,..,,. 5) None

such that (igcity-_Tqo.h,ow mAny ways this

There are 10 boys A- a, i, b, E, F, G, H, l, J iil '**any.wayq

6 boys canselected if E and F do not want to be together? (.5C.,1 ' ,( , : ''

1) 70 2) 140 3) 21p , . 4) 35 ws) 112'' -- ". '1

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Five girls have to be selected out of eight girls A, B, c, D, E, F, G, H such that

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"/ -/Care,e.r Cafd,The simplest way to crack CAT

Hef p line No. 95012-55012Page 9 of 26

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i) lf A is selected, B also has to be selectedii) lf and only if C is selected, D has to be selectdd.In how many ways this setection can be done?1) 18 2) 1e 3)8 4\14 5)12 1 r

;i';5V' A committee of five menrbers is to be formed from 10 persons A, B, C, D, E, F, G, H,

l, J such that the committee shoutd includg exagtly one_persdh-Tiom*-A'dnd B andexactly one peirson from C and D? 2e r *

)c'r 6€ ' 1> :,v -Fn1)20 2)24 3)40-x. 4)60 : 5)80

."J1' | .^..i. La, 3l-r.3cl6A'committee of five members is to be formed from 10 persons,A, j, I, D,,E, F, G, H,53.

, 58.

59.

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4 . .r\,!

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l, J such that the committee should inc[ude at least one person from A and B and at .'

feaSt One persOn frOm C and D? l 'ri'i ''' . '' ?/S 't'bC&r \'r' r 'r"', ;: ,

1) 60 2) 80 3) 140 , . "i'1),1it, **Fl

. ? pr r ,,,, i.

A committee of five members is to be formed from 10 personsb.gl$pff G, H,

l, J such that the committee should include. exactly one persop^from(A and B a4d . r

elaclly-qqe peJga,t-frgm B and c? 4i;i ; i-ii'"' "t'cu"l"\ll-lrii-E "pf "; ?r)

r lTs:-- ifaO .r - 3) 70 2 ' , 4) B0 5) 140 I 'r'11, i:tl; -

t' -ii, ' i'.. ";i''l ,,L t \.

In how many ways five students can be seleited from 6 boys and 4 girls? ifr,t f ,, X'tr)tr )

l?"0:,, ,-T.' 2) 140 3)2?2

i:* { 4) 2s6 Lqk-, .l}y"e of the7fu.,

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|nhowmahyWaySfivestudentscan5d,se|ectedfrom6boysand4girls^suchthatthe number of'boys is greaterthan the number of girls? , t' .l ,-C.y ',C,", ,

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1)60 2)120 '3)180 "4)186 5)252 .r- r,,^"-'6+Q,at OeX €, -EL ( "C.,)/'.1'\

In. how many ways five students cafi. be selected frgr11 6 boys. and 4 girls such thatthereshou|dbeat|edSi2gir|s?.,.).,.''.\t'-.':."'J.tr1) 60 2) 't2o 3) 190- . .4) 186 s) 2s2 .:-' (6xzo) tiqx rt)+ (rxc) .Five students are to be selected from 6 boys and 4 girls sirCh that the numper of girls IShould not be more than 2. In how many ways this seieaion can be done?fi'cr Xtr,,' { "r", Y "Ct,

1)60 .2\120 3)180 4)186 51252 ,1,.

T.e_n hale employees and eight female employees are working in a company. Thecompany decided to send a team of 50% of thb male employees and 25% of the.female employees to abroad. In how many ways this team can be selected? '1.'.' 'fl1)7056 2)252 3) 28 4)720 5) None to.r"L

Sr., ,, ,-z.'zA student has to answer eiqht questions out of twelve in an examination such that hemust answerat least three questions from the first five. The number .of choices .,available to him is: ', ., r tC- +1)35 2) 175 3)210 4) 420 5) None of these s;.C; r15.)"%. to>71 +9739tBS '.-,.', ,"-'\ -'1 ;Four boys and three girls should be selected from 6 boys and 5 girls 3uch that if a V.,particular boy A is selected, his girl friend P also.hds to be selected. In how mana ? Fways this selection can be done? :' ,. r !\r I so 2) 60 3) 1 10 4) 120 5) None i^_

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1) 40 2) 50 3) e0 4) 110 s) 120

A man has 8 friends. In how many ways he can invite 6 or more frienos to his birthday party? -r -- -' - - S6 .,\ r1)28 2)29 3)36 4)37 S)gi. -\ "{.,t, .:t.1! 6J. ri

There are 20 non collinear points on a plane. what is the maximum nffi61," "

,,n"., that can be drawn by joining these points.,1)20, 2)21 ' v3)190

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There are 20 points, out of which 8 are collinear. lf each point is connecied with each,9f th-e otherpoint, what is the total number of different lines tnaf."n bJ;i

,.1) 163 2) 164 3) 190 4)2a 5) None "!Wr8.r* |

r qo_591, z lC aThere are 10 non-coflinear points on a plane. How many triangles can be formedusing these points? "'-"J

S -tp"t* tlU"---' 2)so s)120 4)160 s)ss 'o," ryp

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ffi?il"W,\i:y6o1es c-an be-rormed ,'ro '

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'1 v ue cnosen Trom Inls Dox s_uch that thg balls are of different colours? '9.2t 74+ 842 g5-t 2l-t 2t' 1) 'te t''*u",,,2]1i.*;," 3)66*9'"1 4) 180 ;j N;;; dth;db;;"yffi@,b

,, ",. - l!:t', ila:,+iru:'ff1ft"J"1ir,::g*i:'i' G ; 1 f,,1"#"#'rb,"1o",)"r'-, ilr"i'Tiq:rq5.::::-=::",,riL'" ?lrffi how manv wavs ra.r ! ... a\,:..

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' o'f-1) 675 2) 10s0 -.g),y?5-, , 4) 230_0 . 5) None E') Srtb _ QS. loS* Z B%- ,o4_i;;;L -'0s., L.i::llT l%qgFq1 lll"-rn"ny *.y"

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There are 5 red,4 green and 3 white oltls in a b'ox. tnonSw many ways twoialts can i ,be chosen from this box such that both the bails "r"

pi r"r" "oiJr;;'-

:'" *": 1"" ,. , i ' ': ' 21)19 2)47 <^.,-3)^66 4)1BO --"5)Noneoftheabove .2 -- .

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.,.:i.. 73' 5l::?:5:::*.q"?,1*g$,a box.contains s red,@reen and@elow bals :t73. Five balls are to be ,ciosgn tr.o.m.a box. contains-S red,@reen andelellow balls .\)d:, ,,, ;"n"tntt

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Five balls are to be chosen from a box contains 5 red, 6 green and 4 yellow balls :*'i K '

such that at least two red and two greqn balls shgUd be c

fhic nan ho r{nno? ! | i^,v (,^.7trr ''1 .t f Sr'rx 6--n balls shoud be chosen.Q^.,ztr. l'r .t i 5r^* 6r-l +this can be done?

2) 1200i : rlv Qr..ztr, I .t t 5rax 6 r.) +

3) 800 4) 600 5) 3501) e50

ln. how many ways a cricket team of gleven players can be gelected,from nipe .batsmen and eighi bowlers?1) 12376 2) 221 3) 728 4) 5040 a^ 5) None oflhesg. ;_

In how many ways a cricket team of eteven p;"rrtllrn';"' ;"*i l;;r n n"batsmen and eight bowlers such that the team should contain 6 batsmen and 5bowlers? 4C, x 3c1) 40e6 2) 4704 3) 1237o 4) B: , u) 140 b '

irr:,. , ,.,t.':tn how many ways a cricket team of eleven players cin.'.b" seleCteO irbld niri'e

I

batsmen and eight bowlers such that the team should contairi at most 7 batsmen andat most 5 bowlers?1) 2520 2) 4704 3) 7224 4) 12376 5) None of these

together?1)6! 2)4lx6l 3) 10! 4) 6! x 3! 5)7lx4l

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_ in the .. ,- , ':r'1.,r.tOUrnament? u1\'.'."r'. , .'! .

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J)190 2)210 31 231''o 'i) ido' " 5)Noneofthese '',1 . "jj,,' 7,.lg/ uo* many diagonals can be drawn in a polygon of'10 sides? " i :

..)\/- 1) 55 2) 45 3) 35 ,nrr) - -430 5) 10 . | :

,. 5'/: 1\, a{ tner. are N boys in a class. Eactr pos3iUreidiitot Ooys sings a 2- minute song one

"Y., pair after the other. The total time taken for singing is! | hours. What is ;12.' " | !:

^ :- '

r i 1) 10 2) 11 g) 12 4) 17 el""- 5) llqne qfihese'':' -" ?* 6:4" '"" 'i

Jf lfeLe are 5 chairs in a row. In howrnany ways _5.

persons can sftdn these 5 chairs? i;; .:a ,

' 1) 5' 2)25 - 3) 125 4) 5! 5) None ' .. -/ C*'ffilN 7t9' 51tt ' I<'o^'c -""-1,'a ^2'

3/ In how miny ways 6 bols can stand in a row'? :. : u v

- 1)6 2)36 3)720' 4)466s6 S)None

gy' tn how many ways o oov$Afio i drl, i# ,dan;. in " row? : ,',, :. i

- 1)6! 2)4txpl 3)10! 4)6!x3! 5)7!x3! : ,i. ffg-X t a. .i_r /*r- ! '...,^

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3{ In how many ways"6 bciys and 4 girls can stand in a row such that all the girls stand

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Py In how many ways 6 boys and 4 girls can stand in a row such that no two girls stand\'/' together? '

1) 6! x 840 2)4lx6l 3) 10! 4)6!x35 5)7!x3!(^*- - .1o.. ,1 r, ti ' lr. ,, ..,' i)' i,," :l.t

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tn how many$?5fi'6 mangd dees and4 orange tiees cbn be.ptanted in a row?4) 6! x 3! 5) 7!{ 3!

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Wcsr.f 1 how many ways 6 mango trethat all orange trees are together?

in a row such

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1)7 2)6t 3)10! 4)6!x3! S)Ttx4l

f1 how many ways 6 mango trees and 4 orange.trees can be plantedthat no two orangetrees are together?1) 840 2)-Q5 sl ror 4) 6! x 3! 5) 7! x 3!Gr,)tviy .r* -lIn how many wayd 8 boys can sit around a circular tabfe?1) 6! 2) 2 x7t 3) 8!

in a row suchf , 1tL'i ' lj

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In how many ways B boys and 6 girls can sit around a circular table?:, ,'j,.,:1) 8! x6! 2) e! x 6t 3) 13t 4) 14t si ii,,-5i :In how many ways 8 boys and 6 girls clrn sit around a circular table such that,all girlssittogether? le t,, 1: t ' i.

.|(),':'i'',.''l,}u,|-:|i,',:V.':..,';,i.;...!:''.:i'L.],.,

!,1"Y, !!!y, wlrs I boys and' 6 giirts can sit arolrnd I circular tabte such that no two

1) 6!i 1ii'" ,'.1

girls sit together?1) 13! 2)Bx7l

l -rtlto t) l::'l \3)8ouxg1 a)Qprxt! - 1 51Bo.r6;

93. In how many ways the letters of the word ,MUMBAI, can be a' 1)6!' 2)s! 3)2x5t 4t3x5! sj

94. rn how marly ways.the^retters of the word-'p{Kc4[gfq' can be ananged such thatall vowels are Jogethep : .1..:,- ," . ,,\ l . ,'1)48x61 2)24x61 3)12x6! 4jZl ,St S)6! 6lx,-',, ,

u85' I t?y ry"1 ways the retters of the word 'BANGALoRE'cqn be ananged such that'' no two vowels are together? (i" .r A^f n,u.r,-r-:, ../1) 360 x 6t zl ooo-' Si 3) 180 x 6t 4) 180 x ur

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/ t3u, ln how many w.ays the retters of the *oro 'buebi63, ""n'b"

prr"ng"d such that ail :- r t/ - vowels occupy^the odd positions? tr 'c f, I jI. 1)120 2)240 3)360 4)160 S)Noieofthese

I n .1,' [l-"1 rylv wavs the letters of the word 'eu .NT' can be arranged such that A, .'- -. '.'

| '-/ appears before'U'? -- - -.:-- --"'[ ,izo- - -- - ;iuo s) 120 e1+a

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I r'':. :.,I 'gi !or^,T.nv S.digit numbers can be formed using the digits 1, 2,3,4.5 with oui ,'.,y .,.'\ '=' L'iJ,iiill'J,ini'iliil:';ffi'ffS,i';,'':T:[H:"n the disits 1' 2' 3' 4' s with out '':

I 1) 40 ' ...2) 60 3) 12o ., ojn 'n" tunt

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/9,9How'.1any5'digitnumberscanbeformedusingthedigiis1,2,3,4,.5withoutI repetition such that thetrundreds digitis-moqe th-an the t!n. .iigj] .io i"-n. aigit i,I morethanthe units digit? ( ' '. .,..,ir-) ,,,r ,..,^nr,r...r..) d ."\ \, a.\. \... \L , 1l4o 2)60 s:htu -'-Tzot" ''Tl'ii" )-'-'--'=-i:.

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the function, if a particular guest Gourav wants to give his speech iryEediately beforean other gLlest Sourav? (Lt I €t")

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an other_.9Uest Sourav?1) 120

1) 55 2) 45

The maximum number of1)55 2)45

1)6

1) 60'

. 't)(l'l-" '') 7->+)48 r5) 24 t'

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- t)v 4L: Six guests were invited for a function. In how many ways they can deliver speech an

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! -raaw W2)_ 360 3) 2405+" --/ t

The maximum number of points of intersect

the function, if a particular guest Gourav wants to give his speech before an otherguest Sourav? H b {'.^.", ( I ,; ,, J '.' .tltzo 2)360 'b)zqo 4)48 5)24- ' .,,

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\:Y3: 24tJThere are 5 cities A, B, C, D and E. Each is connected to each of the other city by 3different roads. In how many different ways one can go from A to B?1) 120 2) 486 3) 678 4) 848 5)960 F':t% c 'D

t 103.!

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lf 5 lines and 8 circles are drawn on a plane, what is the maximum numberintersection points? A;^\ + k-vrt \ x51. Elj! " \0 ,i1) 56 2) 66

t&o 8Li%" 4) eo ! , 5) 146 zSix lines are drawn on a plane. No two of them are parallel and no three of thempass through the same point. The number of infinite regions that the plane would bedivided into:1)6 y12

Six liries are drawn on a plane. No two of them are parallel and'no three of thernpass through the same point. The number of regions (including finite and infinite -,'Yiregions) that the plane would be divided into:

2) 12 3) 20 4) 21

2) 66 3) 78

rof ten circles is:

5) 22

. 4)3to l-',h>in 3 boxes such that

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c3 teams of 2 players each are to bg formed from 6 people. The num.ber of ways thiscanbedoneis: '<k, -â*t""b' 3X e' - 5,t6x31) e0 2) 60 3) 120 ,{Ldl, 5) 45

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12 different coloured balls have to be placed into three boxes with 3, 4, and 5 balls. -How many ways this can be done?1) 27720 2) 4620 3) 9240 / . 4) 13860. 5) None , 't _, r: r :

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f n how many ways o.!e or more questio$ iah Oe selected from 10 que$tions? , "1) 10! 4 1014 3) 1023

._ro ,4) 1025 5) 1022 t o.

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How many non nega-tive intege_r lrrp-lejs of a, !, c_satisfy the equation a +1bl+ c = 16? (-1) 153 .T147 q144* 4) 120 5) i.rone ' ,', ,_c

rhe number or -ff n r@rtions or p, ll r ynic1'' ;",tf;rffi*':r + s = 20 is: + zq'; zt2r*2:,rs ' "'- yyvr

2c1) 1140 2) 1771 3) 2024 -3 f q) 4845 5) 969 .{-.

dFindthenumberof positiveintegral solutions or^/oic ,i,o=ro, .r ,* i.1) 1140 2) 1771 3) 2024 ,e,*ri@ni 5) e69 ' ,, -( i ,..

Find the number of disjinqlterms in the expansion oii" * U + C I d)2s? t' -

": " i,,1) 1140 2) 1771 3) 3276 qiTaist - -6)

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No'nfi{&lvt"r ; h"rht3)*'EIn how many ways 20 identical chocoulates can be distributJt to 5 children such that

lf a + b + c = 20, a> 1, b > 2, c >1) 153 2) 147 3) 144 4) 120 5) None

eachgetsatleastonechocolate? l''.ir ff*'n ',r,-r:,.'' i"t,"..it;-'],'. ., .,-,

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lf alf the four digit numbers that can be formed using the digits 1,2,i ,4,5 withtout orepetition a/e written in ascending order, what is the 50* number?

2) 31254 232145

aI lS Ine bU"' number? | _ pr.,4) 3215Af:' 5) 35421 . ',

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1) 11255 2) 11311 3) 12311 4) 123sE 5) None

Exercise ' )

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A new flag is to be designed with seven vertical stripes using some or all of thecolours-Orange, Green, Yellow such that no two adjacent stripes have the samecolour. Then the number of ways this can be done is:1) 48. 2) 60 3) 96 4) 192 5) None

l_o* many positive odd numbers less than 105 can be formed using the digits 0, 1 , '4,

5?r.1) 256 2) 512 3) 384 4) 768 5) 1024

How many numbers greater than 500 and less than 5000 can be made using thedigits 0, 1, 3,4,5, if the repetition of the digits is allowed?1) 375 2) 4oo 3) 399 4) 401 5) None

ff 4000 < x < 12000 and x cannot have'any digit other than 0,2,3, 5, B. How manyintegers x can take?1) 250 2) 309 3) 310 4) s11 5) None

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A six digit number is to be formed using some or all of the digits O, 2, 3, 4, 5. Hcwmany such numbers can be formed such that no consecutive digits of the number are

7 i 'N' candidates appeared for gn interview. The interviewer has-to_select _at$* a ,l {r:I candidates.'The number of different choices available to him is.{!. Find 'N'? ZT-ll1).4,., '2)5 3)6. q! q8(.r ,.20-1=g .,:.| 'r'),;=

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,br) )ri"*. t :T.,dl9*es out of '2n+.1',ca:dr*\es. The number of. choices available to him is(

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''/l 12,7 1 1\2 2)3 3)4 4)8 5)9 16-,.,c.,' vl ,:"e--9. - Let S be the set of all six digit numbers that can be formed using the digits 1, 3, 4, 5,

7, 8 with out repetition. What is the sum of the ri Itof the numbers in S?1) 120 2) 720 3) 3360 4) 5)'None

A man has 8 friends: 4 boys and 4 girls. ln how many ways can he invite 6 of lhem ,

forapartysuchthatthenumberof boysismorethanthenumberof gifls?l\gux3) * -ul'1) 4 2) 6 3) 12 4) 28 5) None 2"

,21 Vineel has 11 friends: 5 boys and 6 girls. In how many ways can he jnvitg ! of then . .. r,- f.ora party if there should be atleast 4 girls in the invitees? tuX't t3 .J r. ''f"1) 60 2)75 3) 135 4)'t45 5)270 "c, x t, )

12. Vikranth has 6 friends: 3 boys and 3 girls. His wife also has 6 friends: 3 boys arid 3 'girls. In how many ways can they invite 6 of them (3 boys and 3 girls) for their( lS\ \ )+wedding anniversary such that 3 of them should be Vikranth friends and the-remaining three must be his wife's friends. t (lx

I

1) 144 2) 162 3) 164 4) 81 5) 82

13. A team of 5 members is to be selected from 10 members A,

Câ :\ :r A :^ ^^r^^r^.t r-r ^r-^ L^^ r, i) if A is selected B also has tb be'selected t' ';;

ii) exactly two have to besdleOted from D, E, F | ' : , ., -l

iii) if and only if C is selected H has to be Selected :

, In how many ways the team can be selected?1)'l'l 2) 12 3) 33 4) 36 5) None of these

'; I ? '''' '":, 14. . In how many ways six persons Sivaji, Gandhi, Vamsi, Grahambell, Denmark and'*+-

5 Bose be seated at a round table, if Sivaji has to sit between Vamsi and Gandhi? .r.'' '- 1) 6 2) 12 3) 18 4) 36 5) NoneI iV ro-,, ,) i'r'.' li V.'n'<, . ,.' ,., 1.i'., r'.lf ''

5ll Z - lZ15. How many 5 digit numbers can be made using the digits 2 or 3 such that the digit 2

cannot be in the adjacent positions?1) 6 2)8 4) 13 5) 26

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How many five digjitunufrrbers can be formed using the digits 1, 2; 3, 4" S.rnrith out ,,_,.repetition suchthatexactlytwo odd positions areoccupied !VoOO digits? SxZX3 yZXIX A)1)72 2)36 3) 48 4)24 Cp^^+uuu,JJi^0, -to :,,h";'; nr.,ff

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How.many five digit numbers can be formed using th'e digits 1, Z g, A, S with outrepetition such that exactly two odd positions are occupied by even digits? {1,.,1)72 2)36 3)4s 4)24 s)144 4-i--r21r.-xqxr

(There are 10 non collinear points in a plane. lf each point is to be connected with at'l v1.

t least one of the other point, what is the minimum number of lines that has to be Ii drpqg? .f.\ |

!rlU*, 2) 10 3) 11 4)4s 5) 5s Ky I

I There are 10 non collinear points in a plane. lf each poini is to be bonhectea.*ittr "t{\ least one of the other point, what is the maximum number of lines that ian bel

I drawn? )U) 9 2) 10 3) i 1 4) 4s ,;,,i,. t5) 55

How many four digit numbers can be formed using the drstts g, 4, F".O,. g such that "r.-' i.;i^ ''

thenumbershou|dnotchangeevenifthedigitsarereversed?tjt..lLl..J+.'|..l---,1) 20 2) 40 3) 400 4) 25 5) 625

In how many ways five persons can be selected out of 8 couples such that thereshould be no couple?1) 1792 2) 6720 3) 21s040 4) 4368 5) None of these':

In how many ways four numbers can be,:chosen from the first 10 natural numberssuch that no two numbers are consecutive?1) 1e20 2) 210 3) 35 4) 1001 5) 330

.23. A box.contains g4ggkal ied balls and I identical green balls. In how many ways t?c-five balls can ber4fi6.@Fom this box? ' lr l- .', I '1)6 2)E- 3)131 4)1287 s)Noneofthese Ill a-,''.''_

, t'3bThere are 10 n0angoqs, 12 apples and 15 oranges in a basket. In how many ways sixfruits."n b@gedlrom the basket?1) 28 3) 21 4) 3u 5) 6.

lf 'all the four digit numbers that can be formed using the digits O, 1 ,3,4,5 with outrepetition are written in ascending order, what is the 60th number? \

1) 4135 2) 4351 3) 4531 4) 4153 5) 4513

26. lf all the four digit numbers that can be formed using the digits 0, 1 ,3,4,5 (repetitionis not allowed) are written in descending order, what is the 50th number?1) 3045 2) 3450 3) 3504 4) 3054 5) 3540

What is the sum of all the four digit even numbers that can be formed using the digits1, 3, 4, 6 with out repetition?1 ) 4056 2) 4096 3) 9112 4) 9332 5) None of these

Cars,er Cafd,The simplest way to crack CAT

Help fine No. 95012-55012Page 17 of26

18.

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6A There are six chairs numbered 1 to 6. six boys have to be seated '.1,t'.{S"io . r'i

-..^ such that a particular boy doesn't want to-sit glrja prime numbered chairlnd ai6tner d*ctf 'qF\Y F' boy wants tb siton even numbered chair?)?'"Ql r L\ 5) 1 I ltl " \7' ...i 1) 144 2) 168 3) 216 4)72 5) 96

@ APsRrc has started,a new bus jfm !y-d:I?b:d.lo-.Bfls"1"-r-"-:!."1-tnat ittras tg,r63)9-J t.r ,srops In between. How many differeni tickets that the conductor should naGZ

. i;\ ; .(AFume that a ticket from statior2fl to B is same as the ticket from station B to A))'l' ,rr ,.r o\ or\ at al lrz<f i\ qtr tr\ ant l. "'llr-)* ?'lro ;!P,'. ''^ i{ ,r, \ocs 7:,?, ,

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' arl -The figurc below shows the plan of a town. The streels are at righi #gles to each other. A qr> '

Orectangular park'P' is situated inside the town.with a diagnol roadliiunning through it. There f"" ,is also a prohibited,region 'D' in the town. .n\ ?

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| )*t , -lvLNeelam rides her bicycle from hei hpuse at 'A' to her office at 'B', taking shortestpath. Then the number of shortest paths that she can choose ls:1) 60 2) 75 3) 4s 4) e0 5) 72

(cAT 2008).

Neelam rides her bicycle from her house at 'A' to her club at 'C', via '8" takingshortest pgtl\ Then the number of shortest paths that she can choose is:

3) 7e2 4) 1200 5) es6

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rrv'!'1" t-crrr ue ,t , _,., .1).fn 2)ry:-j,ffia 5)501

Jrr- Wnl^': the number of distinct^terms in the expansion of (a + b + c)mz ) r.,,,' ' 1) 231 2)2s3 3)242 - -6;:;; q22B(cAr 2008)

\" l9 IlT l"6 tasks and 6 persons. Taskl cannot be assigned either to person l,r.oito ,r, , . ' ' -. ,'' person 2'Tast<2 must be assigned eitherro p"noni^*-ro pil;;:-Ei6;Lyn,. . .to be assigried one task. In ho-w many way, [r" ini. asstgnment be done?', 1) 144 2) 18o

, . , 3),tep' -,-;iido- 5) 216''-

, ( :j:--' - "' '::-"' (cAr2oo6) .,,rr l'.'{ tn a chess competitio"l:y:1."j"_q

::r:,boys.and girts of a schoot, every student had I c.: .r,.}.. . r ,to pray exacfly one game. with every othei studeni. riwas round that in 45 gamesboth the prayers were girrs.and in t'so g.rlr bol'; were ooys..the number of games,l yl,gh one ptayer wai a boy and tt"iir,"i-pr"v"r,1) 2oo " ,iri;- 3) 23s o) ,uu '"t

a sirt is : I orr' , {(cAT 2005)

Let s be the set of five digit numbers forme-d by fhe digits 1.2,3,4,susing each digitexactly once such that exactly two odd positions ar" oi.rpied by odd digits. what isli"il!.'of

the disits ilT,l,tht

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N persons stand on the circumference of a circle at distinct points. Each possible pairof persons, not standing next to each oiner, sings a two-mrnute song one pqir afterthe other. lf the total time taken forsinging ir 26-minutes, what is N? 'lL1 ,i:1) 5 2)7 3) g v ' .-o) None

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i:,T::lf:l,igj,,.n^:'";,T^,'f:.,:T:::ll9ng *"y roads arrowins travel onry northl;1:!?^9r

west wards' Along how many distinct rouies ."n "'."";;"";;#i':[,'19

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Ca reer Cafd,cAT * MAT. csAT * GRE *GMAT

(cAT zootl10' A new ffag with six vertical.stripes is to be designed using some or afr of the corours

isi4;lliil;3 xiit*jihlil$J'iorwai,ini, ",; oe oone J; that no two1) 12x61 416rig'z 3)zoi125 4)24x216(cAT 2004r'

Questions 11and12:A string of three english letters is formed(i) The first tetter is any vowl;-'

" 'sv as per the following rules.(ii) The seconrr r.++^, ;l - -- "'(iii) rrthi:^":ldftlgrisF' norp A'!''"t. ,r4 'f^\xt z >',

,r"r.,i!l?ll tetter is m, then the third l"tre, i" any vowe, *,.n n different(iv) ,, ,n:

::::"q tefter is n, then the th;rd tetter is either e Lr u , r.j" , . ZO

;:"-^'^'^::;il:,::::.'**ffi T-retteri'ss-am-e"*1,,^,,5'1i,.".-ui' 1)40

. u" rx2if ;;' ;' ? 5P"'" :

o"too"Tu"o usins theabove rules?'----o ( v

.roi,_. V ,qr5*i ' t^txt' (cAT 2oo3), rz. now many strings of letters can possiblv Ithe third letter or tne string is e? f ,le ,t",,t_T?f ,l.iXo the above rutes such thaf ;(1)8 (2)g --"''"ijl 10 ",orrr''I r-r '' : rrrji,._r ,-r

I 13. There are 12 towns .r2.,,,^^,.r ;-r^ , (cAT 2003)

{., *;;;l;l?::'.Tji:rfis',iifri:,$#ffi"Ji::j:il;::j#jifi:fi;:[

"r,,tlffioure'wis-el il'"':'i':-'': 11 thev belons to the

(1) 72 ' '-wrnanv drrect'?l"rogt" ,'nu. ,0, ,oo \

, ,_ (cAT ?003I t,@agraph may be defin"g

": a set of points connectel gv l,r"', ;,uo "on"s.

trveryedge connects a pair or poinis' inri a.triangr" i, , gr"ph with 3 "og",

and 3 points.The desrgg of'poiniit th;;;o"ili'"og"r;onn"It"o

io it. roi"i"rpre, a tria,rbfe isa graph with three points. tg"g;J u3ln. conrlo"r a graph with 12 points. rt isffii3:"?"T;:l T 11,",': ffi *gH J,;,ilj'g;:,1

€ s e q u e n ce o r e d g e s rh e1)11se<66 2).oa;s,ff 3)11<e<65 4)o<e<11,1 (cAT zooq . r

,,rc.ff1, t12, l.l3 .... n1e ?r€ 10 numbers such. that n1 > 0 and the numbers are given in\'/ ascending order' How t3ny tripfets can oe fo#ed rlsing these nuro"r, such that in .:T:iif;:"i ,l[:,flfi;Xil|:;; H' ti"n tn" s",ondiumber, ano ine second nu::rber1) 1oe 2) 27 q T2o 4) po toca

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16'ffow rirany numbers between 0 and one miilion can be fo'ned using 0, 7 and.g?

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.,.'r, .17. ln how many ways, we can choose a black and a white square on a' ' o such that the two are not in same row or column?r,,"'.r/ 1) 32 ' 2) 96 3) 24 4) 768

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. .- Questions 2O and 21: --1I '-' '' ' u .r -. X: ' n 1,

9\(:.. )/ . f t,..,...There are 11 alphabets A, H, lr M, O, T, U, V, W" X, Z appears same when looked ina.mirror. They are called symmetric letters. Other alphabets are called asymmetric )a

:letters. - -ii .. I f" t I':'''\- :

Jpi How many four-lettered passwords can be formed by using symmetrical letters onty?(repetitions not allowed) , , ''-'

1) 1086 2) 255 3) 7920 4) None,of these(cAT 20021

How many three- lette.red vqofds can be formed such that at teast one syn'tmetrical - ,

tettef is thefg? ^r,_e trf,i t;r ,- , ).. ;^.| 1 i . ,., 11,.1 t r' ,.r-: t -'

1)12870 2)18330 3)16420 4)Noneofthese ! i 'i.r"; --'-' r*;{F€ 'v r-v '"-

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-'''"'i- (fiOnured flag, three white flags and two blue flags are ananged in a line "r"n',n;r,'tyZX\x -- (l) no two adjacent flags are oJ the.same colour pur a r.,7g,_. I _O

' ' I \, (B) the flags at the two ends of the line are of different colours. ;: ; iq1 g -. inhowma-nydifferentwayscantheflagsbearanged?'l:.- 3w -:-x-

i+. l 6 2)4 3) 10 4)2 tîl'r' t ', 'i . , j .. :,, (CAT 2000) Bw a I.r r N *rl

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(Zi; Sam has forgotten his friend's seven :digit tetephone luFLeLl.|g_tennembet1the *_-_-s\'-.,/fc||owingthefirstthreedigitsareeither6Q5oio7+,@-'non".-,.,|number nine appears once. lf Sam were to use a trial and error process to reach hisfriend, what is the minimum number of trials he has to make before he can be certainto succeed?1) 1000 3) 3402 4) 3ooo {x 2x 2-

(cAT 2000)2) 2430

07.tn re are three cities A, B and C, each of these cities is connected with the other tvro\'" cities by at least one direct road. lf a traveller wants to go from one city (origin) toanother city (destination), she can do so either by traversing a road connecting thetwo cities directly or by traversing two roads, the first connecting the origin to the third A. ' .., / ,

. city and the second connecting the third city to the destination. tn all th'ere are 33 :.routes fronr A to B (including those via C). Similarly there are 23 routes from B to C(inc|udingViaA).HowmanyroadsaretherefromAtoCdirect|y?|_Lilo 2)3 3)5 4)10 r

AIB(cAT 2000)

2a + ;* '23. Ten points are marked on a straight line and 1 1 points are marked on another I + ?t 'l

straight line. How many triangles can be constructed with vertices from among theabove points?

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, numoet oI QlIIgIeIlt ways (it st:tgl;u()tl9t al rtrdsr u||ir ucr rqrvarg 19 (,r, lrrrr ||ro^rlurrr- / P)T- :1' ... ngmber of candidates ihat cd-Fb6-sercaed for the scholarship is -r,. . rr( 1l'

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n.";:--fz 2)4 3) 6 4)5 '? .:h. .izl'i'i..'. (CAT 1999) ,r.'.,^Lrl ''." 25-. How many numbers can be formed fro m 1,2, 3,4,5 (without repetition), when lhe- JJ- 1n-- )

u,/ ai.gn attne unit's place must be greater than that in the ten's place? *'-.-._. 7f V, )L" ".iif" - - t)'60. 3) 40 4)2x 4t '.-'!

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,',,-2b,)|nhowmanywayscaneightdirectors,Vice-chairmanandbhairma4ofafirmbe' t - " seated dt a iound table, if the Chairmqn hAs to sit between the Vice-chairman and a..^4i'ri&on 9*9 vcc D::.. -

r/. 1+vc z) rrv r) rv"iJ -t ' (cAT lggg) ( 2^r r) 1 a ( 1

Q},irora scholarrnip,@tn" andidates out ot?L1ldan b",l1gslpd. ffite i"^'l:-b*-s#'.nu'berofdifferent@ionofat|eastoriilclandidateis-63,themaximunf}.tt

(;"ni:', 2)2xat 3)2x7t f .+;tton"orthes8 I v-q u' -\,c6sr,"* lGar tsszl ^

rcxjU;'I'9rt27 . A man has 9 friends : 4 boys and 5 girls. In how many ways can he invite-them;f'-JjS\

therehavetobeexactly3girlsintheinvitees?___'r.., q!., ('rp--,1) 320 2) 160 3) 80 .-, 4) 200 r.--

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f28.)A five-digit number is formed using digits 't , 3; 5,7 and 9 without repeating any onevof them.--What is the sum of all such possiuie numbersz 5 y'Y )Y;rYl - '':oX Af

1) 6666600 2) 6666660 3) 6666666 r',4) None of these - i

. tt, ,, .'. : -.'' . Jl.'.i.,',' .i''t."r'r:'l ' ' ,".,JQAT1993) J"*""

.i'--.r,? \ u ..:i<,r- ?-.r/t\lllll{.;2gfihow many ways can the tetters of the word ABAbUS be rearranged such-tha&the-:-;'

\/' vowelsalwaysappeartogether? RCg (Anr;)1) 6lt2l .2) 3!*3! 3) (3!.3!y2! 4) @rc\l2l ' '/ r I

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lt3::" '. q'1g0. A rive-digit number divisible by 3 is to be formed using numerical "q,i'; .i, s,4 and 5'1"'?a

withoutrepetition.Thetotal numberofwaysthiscanbedoneis: lrttt!r, tfXSc{Xaf 21) 122 2) 210 3) 216 4) 217 I i'

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3J^tfhere are 1 0 stations on a railway hne. The nurnber of different journey tickets that

,,./ are required by the authorities is : " .r..;..--:..: - : : -1) 1ol 2) 90 3) 81 4) 10 \ri i,.

. , (SNAP 2008) ri '{'1.,d.,r'.'(, . i.rci '\,'.,; i' r '.tl '/

fiA number lock consists of 3 rings each marked with 10 different numbers. In how,./ many.cases the locks cannot be opened? . - | ^ cq1r)s't 2)103 sl sd "/i)sgs lg\se - l" " I

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33. Six straight lines are drawn irY a plane with no two parhllel and no three boncurrent.The number of regions into which they divided the plane is: 1l(7 rr) * 1

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-/\ rthe vice-Chancellor of University of Delhi decided to form a committee to look intov the feasibility of introduction of semester systems at the under-graduate ievel in the

University. Q members from the Executive Council and 7 memb-ers of the AcademicCouncil were found suitable for the job. In how many ways can the Vice-Chancellorform a committee of O members such that at leasi 4 members of the committee

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36.rvhexumberof distinctterms in the expression of fi | y t zlW)3o are: ) ^/ 1) 4060 . ?), s456 i 3) 2740s 4) 46376' " i ,L?, ,,\/ ,11 ,-- = 3rX3-xaEr' (llFT2gogl " "':'l i :,', - --- A ', .,h.

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Permutations and Combinations