Mathematics

PERMUTATION & COMBINATION Practice Sheet

1. How many numbers divisible by 5 and lying between 9. In a train five seats are vacant, then how many ways
3000 and 4000 can be formed from the digits 1, 2, 3, can three passengers sit

(a) 6P2 (b) 5P2

:
4, 5, 6 (repetition is not allowed)

y
vad 1, 2, 3, 4, 5, 6 ls 5 ls foHkkT; rFkk

b
(c) 4P2
3000 ls 4000 ds chp

(d) 6P3
,d jsyxkM+h esa ik¡p lhVsa •kyh gSa] rks rhu ;k=kh fdrus rjhdks
ldrs gSa\
fdruh la[;k,¡ cukbZ tk ldrh gSa\ (iqujko`fÙk dh vuqefr ugha gS)
(a) 20 (b) 30 (c) 10 (d) 60
10. The sum of all 4 digit numbers that can be formed by

s
using the digits 2, 4, 6, 8 (repetition of digits not
2. The number of ways in which 6 rings can be worn on

c
allowed) is

i
the four fingers of one hand is
vad 2, 4, 6, 8 (vadksa dh iqujko`fÙk dh vuqefr ugha gS) dk mi;ksx

3.
(a) 46

a
6

t
(b) C4 (c) 6

S
4

r
,d gkFk dh pkj vaxqfy;ksa ij 6 vaxwfB;ka iguus ds rjhdksa dh la[;k gS

i (d) N.O.T
There are 3 candidates for a post and one is to be
djds cukbZ tk ldus okyh lHkh
4 vadksa dh la[;kvksa dk ;ksx gS
(a) 133320 (b) 533280 (c) 53328 (d) N.O.T

m
11. The number of 3 digit odd numbers, that can be

a
selected by the votes of 7 men. The number of ways formed by using the digits 1, 2, 3, 4, 5, 6 when the

e l
in which votes can be given is repetition is allowed, is

h s
,d in ds fy, 3 mEehnokj gSa vkSj muesa ls ,d7dk
iq#"kksa
p;u ds 3 vadksa okyh fo"ke la[;kvksa dh la[;k]1,tks2,vad
3, 4, 5, 6

(a) 73

a t ai
oksVksa ls fd;k tkuk gSA oksV nsus ds rjhdksa dh la[;k gS%dk mi;ksx djds cukbZ tk ldrh gS] tc iqujko`fÙk dh vuqefr gks] gS
(b) 37 (c) 7C3 (d) N.O.T (a) 60 (b) 108 (c) 36 (d) 30

M raj B
4. If nP4 : nP5 = 1:2, then n = 12. How many numbers of five digits can be formed from
the numbers 2, 0, 4, 3, 8 when repetition of digits is
;fn nP4 : nP5 = 1:2, rksn =
not allowed
(a) 4 (b) 5 (c) 6 (d) 7
la[;k 2, 0, 4, 3, 8 ls ik¡p vadksa dh fdruh la[;k,¡ cukbZ tk ldrh
5. If nP5 = 9 × n–1
P4, then the value of n is
gSa] tc vadksa dh iqujko`fÙk dh vuqefr u gks

e
;fn nP5 = 9 × n–1P4, rksn dk eku gS (a) 96 (b) 120 (c) 144 (d) 14

e
(a) 6 (b) 8 (c) 5 (d) 9 13. The numbers of arrangements of the letters of the
6. The value of nPr is equal to word SALOON, if the two O's do not come together, is

7.
n

N
Pr dk
n–1
eku cjkcj gS
(a) Pr + r Pr–1 n–1

(c) n(n–1Pr + n–1Pr–1)

n–1
(b) n. Pr + Pr–1
(d) n–1Pr–1 + n–1Pr
n–1

There are 4 parcels and 5 post-offices. In how many

different ways the registration of parcel can be made
;gk¡4 iklZy vkSj
5 Mkd?kj gSaA iklZy dk iathdj.k fdrus vyx&vyx
'kCnSALOON ds v{kjksa dh O;oLFkk dh la[;k]O;fn
ugha vkrs gSa] rks gS
(a) 360 (b) 720 (c) 240 (d) 120
,d nks

14. In how many ways n books can be arranged in a row

so that two specified books are not together
lkFk

fdrus rjhdksanlsiqLrdksa dks ,d iafDr esa O;ofLFkr fd;k tk ldrk

gS rkfd nks fufnZ"V iqLrdsa ,d lkFk u gksa
rjhdksa ls fd;k tk ldrk gS\
(a) n! – (n – 2)! (b) (n – 1)!(n – 2)
(a) 20 (b) 45 (c) 54 (d) 54 – 45
(c) n! – 2(n – 1) (d) (n – 2)n!
8. In how many ways can 5 prizes be distributed among 15. Numbers greater than 1000 but not greater than 4000
four students when every student can take one or which can be formed with the digits 0, 1, 2, 3, 4
more prizes (repetition of digits is allowed),
pkj Nk=kksa ds
5 iqjLdkj
chp fdrus rjhdksa ls forfjr fd, tk ldrs gSa] 1000 ls vf/d ysfdu 4000 ls vf/d ugha la[;k,a tks0, 1, 2,
tc çR;sd Nk=k ,d ;k vf/d iqjLdkj ys ldrk gS\ 3, 4 vadksa ls cukbZ tk ldrh gSa (vadksa dh iqujko`fÙk dh vuqefr
(a) 1024 (b) 625 (c) 120 (d) 600 (a) 350 (b) 375 (c) 450 (d) 576
16. The number of numbers that can be formed with the 'EAMCET' 'kCn ds lHkh v{kjksa dks gj laHko rjhds ls O;ofLFkr
help of the digits 1, 2, 3, 4, 3, 2, 1 so that odd digits fd;k x;k gSA ,slh O;oLFkkvksa dh la[;k fdruh gS ftuesa nks Loj ,
always occupy odd places, is
nwljs ds cxy esa ugha gSa\
vadksa
1, 2, 3, 4, 3, 2, 1 dh lgk;rk ls ,slh la[;kvksa dh la[;k (a) 360 (b) 114 (c) 72 (d) 54
fdruh gS] ftlesa fo"ke vad lnSo fo"ke LFkkuksa ij gksa\ 23. The total number of permutations of the letters of the
(a) 24 (b) 18 (c) 12 (d) 30 word "BANANA" is

:
17. How many words can be formed with the letters of "BANANA" 'kCn ds v{kjksa ds Øep;ksa dh dqy la[;k gS

y
the word MATHEMATICS by rearranging them (a) 60 (b) 120 (c) 720 (d) 24
MATHEMATICS 'kCn ds v{kjksa dks iquO;ZofLFkr djds fdrus 'kCn

b
24. How many numbers greater than 24000 can be formed
cuk, tk ldrs gSa\ by using digits 1, 2, 3, 4, 5 when no digit is repeated

s
vad 1, 2, 3, 4, 5 dk mi;ksx djds24000 ls cM+h fdruh la[;k,¡

c
11! 11! 11!
(a) (b) (c) (d) 11! cukbZ tk ldrh gSa] tc dksbZ vad nksgjk;k u x;k g

i
2!2! 2! 2!2!2!

t r
(a) 36 (b) 60 (c) 84 (d) 120

i
18. The number of arrangements of the letters of the word

a
25. How many numbers greater than hundred and
CALCUTTA

S
divisible by 5 can be made from the digits 3, 4, 5, 6, if
'kCnCALCUTTA ds v{kjksa dh O;oLFkk dh la[;k

m
no digit is repeated

a
vad 3, 4, 5, 6 ls lkS ls cM+h vkSj
5 ls foHkkT; fdruh la[;k,¡ cukbZ

e
(a) 2520 (b) 5040 (c) 10,080 (d) 40,320

th i l
19. In a circus there are ten cages for accommodating

s
ten animals. Out of these four cages are so small that
five out of 10 animals cannot enter into them. In how

a a
many ways will it be possible to accommodate ten
tk ldrh gSa] ;fn dksbZ vad nksgjk;k ugha x;k gS
(a) 6 (b) 12 (c) 24 (d) 30
26. The number of 4 digit numbers that can be formed
from the digits 0, 1, 2, 3, 4, 5, 6, 7 so that each number

M raj B
animals in these ten cages contain digit 1 is
,d ldZl esa nl tkuojksa ds fy, nl fiatjs gSaA buesa ls pkj fiatjs brus
vad 0, 1, 2, 3, 4, 5, 6, 7 ls cukbZ tk ldus okyh
4 vadksa dh
NksVs gSa fd nl esa ls ik¡p tkuoj muesa ços'k ugha dj ldrsA bu nl ftlesa çR;sd la[;k esa vad
la[;k] 1 gks] gS
fiatjksa10
esatkuojksa dks j•uk fdrus rjhdksa ls laHko gksxk\ (a) 1225 (b) 1252 (c) 1522 (d) 480

e
(a) 66400 (b) 86400 (c) 96400 (d) N.O.T 27. The number of arrangements of the letters of the word

e
20. How many words can be made from the letters of the BANANA in which two N's do not appear adjacently is
word COMMITTEE BANANA'kCn ds
v{kjksa dh O;oLFkk dh la[;k N
ftlesa
vklUu
nks

(a)
N
COMMITTEE 'kCn ds v{kjksa ls fdrus 'kCn cuk, tk ldrs gSa\ :i ls çdV ugha gksrs gSa

9!
 2!
2
(b)
9!
 2!
3
(c)
9!
2!

21. How many numbers can be made with the digits 3, 4,

5, 6, 7, 8 lying between 3000 and 4000 which are
(d) 9!
(a) 40 (b) 60 (c) 80 (d) 100
28. The number of ways in which 5 boys and 3 girls can
be seated in a row so that each girl in between two boys
5 yM+dksa 3vkSj
yM+fd;ksa dks ,d iafDr esa bl çdkj cSBkus ds rjhdks
dh la[;k fd çR;sd yM+dh nks yM+dksa ds chp esa gks
divisible by 5 while repetition of any digit is not allowed (a) 2880 (b) 1880 (c) 3800 (d) 2800
in any number
29. Eleven books consisting of 5 Mathematics, 4 Physics
3000 vkSj4000 ds chp ds vad 3, 4, 5, 6, 7, 8 ls fdruh and 2 Chemistry are placed on a shelf. The number
la[;k,¡ cukbZ tk ldrh gSa5 tks
ls foHkkT; gksa] tcfd fdlh Hkh la[;k of possible ways of arranging them on the assumption
esa fdlh Hkh vad dh iqujko`fÙk dh vuqefr ugha gS that the books of the same subject are all together is

(a) 60 (c) 12 (c) 120 (d) 24 ,d 'ksYiQ ij X;kjg fdrkcsa j•h gSa ftuesa 5 xf.kr] 4 HkkSfrdh vkSj
22. All the letters of the word 'EAMCET' are arranged in jlk;u foKku dh gSaA ;g ekurs gq, fd ,d gh fo"k; dh lHkh fdrkcsa
all possible ways. The number of such arrangements ,d lkFk gSa] mUgsa O;ofLFkr djus ds laHkkfor rjhdksa dh la[;k D
in which two vowels are not adjacent to each other is (a) 4!2! (b) 11! (c) 5!4!3!2! (d) N.O.T
30. The number of words that can be formed out of the 36. n gentlemen can be made to sit on a round table in
letters of the word ARTICLE so that the vowels occupy n lTtuksa dks ,d xksy est ij cSBk;k tk ldrk gS
even places is
'kCnARTICLE ds v{kjksa ls cuk, tk ldus okys 'kCnksa dh la[;k bl(a) 1 (n + 1)! ways/rjhd ls
çdkj gS fd Loj le LFkkuksa ij vk,¡ 2
(a) 36 (b) 574 (c) 144 (d) 754 (b) (n – 1! ways/rjhd ls

:
31. If the letters of the word SACHIN arranged in all
1
possible ways and these words are written out as in (c) (n + 1)! ways/rjhd ls

y
dictionary, then the word SACHIN appears at serial 2

b
number (d) (n + 1)! ways/rjhd ls
;fn SACHIN 'kCn ds v{kjksa dks lHkh laHko rjhdksa ls O;ofLFkr fd;k

s
37. The number of ways in which 5 male and 2 female
tk, vkSj bu 'kCnksa dks 'kCndks'k ds vuqlkj fy•k tk,] rks
SACHIN members of a committee can be seated around a round

(a) 603

ti c
'kCn Øe la[;k ij vkrk gS
(b) 602

a ir
(c) 601 (d) 600
32. Let the eleven letters A, B .....,K denote an arbitrary
table so that the two female are not seated together is
fdlh lfefr ds 5 iq#"k vkSj
2 efgyk lnL;ksa dks ,d xksy est ds
pkjksa vksj bl çdkj cSBkus ds rjhdksa dh la[;k fdruh gS fd nks efg

S lnL; ,d lkFk u cSBsa\

permutation of the integers (1, 2,.....11), then

m
(A – 1)(B – 2)(C – 3).....(K – 11)

a
(a) 480 (b) 600 (c) 720 (d) 840

e l
eku yhft, X;kjg v{kjA, B .....,K iw.kkZadksa
(1, 2,.....11) ds ,d 38. In how many ways 7 men and 7 women can be seated

h s
LosPN Øep; dks n'kkZrs(AgSa] rks– 2)(C – 3).....(K – 11)
– 1)(B around a round table such that no two women can

a t ai
(a) Necessarily zero/vfuok;Z :i ls 'kwU;
(b) Always odd/ges'kk fo"ke
sit together
7 iq#"kksa7vkSj
efgykvksa dks ,d xksy est ds pkjksa vksj fdrus rjhdksa

M raj B
(c) Always even/ges'kk le ls cSBk;k tk ldrk gS fd dksbZ Hkh nks efgyk,a ,d lkFk ugha cSB ldr
(d) None of these/bueas ls dksbZ ugha (a) (7!)2 (b) 7! × 6! (c) (6!)2 (d) 7!
33. In how many ways can 5 boys and 5 girls sit in a 39. The number of circular permutations of n different
circle so that no two boys sit together objects is

e
5 yM+ds vkSj5 yM+fd;k¡ ,d o`Ùk esa fdrus rjhdksa ls cSB ldrs gSa
n fdfofHkUu oLrqvksa ds o`Ùkh; Øep;ksa dh la[;k gS

e
dksbZ Hkh nks yM+ds ,d lkFk u cSBsa\ (a) n! (b) n (c) (n – 2)! (d) (n – 1)!
40. A man has 7 friends. In how many ways he can invite

N
5! 5!
(a) 5! × 5! (b) 4! × 5! (c) (d) N.O.T one or more of them for a tea party
2
,d vkneh ds 7 nksLr gSaA og muesa ls ,d ;k vf/d dks pk; ikVhZ
34. In how many ways a garland can be made from exactly
10 flowers ds fy, fdrus rjhdksa ls vkeaf=kr dj ldrk gS\
Bhd 10 iQwyksa ls ,d ekyk fdrus rjhdksa ls cukbZ tk ldrh gS\ (a) 128 (b) 256 (c) 127 (d) N.O.T
15 15
41. If C3r = Cr+3, then the value of r is
9!
(a) 10! (b) 9! (c) 2(9!) (d) ;fn 15C3r = 15Cr+3, rksr dk eku gS
2
(a) 3 (b) 4 (c) 5 (d) 8
35. 20 persons are invited for a party. In how many
different ways can they and the host be seated at a 47
5

circular table, if the two particular persons are to be 42. C 4   52–r C3 

r 1
seated on either side of the host
(a) 47C6 (b) 52C5 (c) 52C4 (d) N.O.T
20 O;fDr;ksa dks ,d ikVhZ ds fy, vkeaf=kr fd;k x;k gSA ;fn nks nfo'ks"k n
43. Cr ÷ Cr–1 =
O;fDr;ksa dks estcku ds nksuksa vksj cSBk;k tk,] rks mUgsa vkSj estcku dks
,d xksykdkj est ij fdrus vyx&vyx rjhdksa ls cSBk;k tk ldrk gS\ (a) n – r (b)
n  r –1
(c)
n – r 1
(d)
n – r –1
(a) 20! (b) 2.18! (c) 18! (d) N.O.T r r r r
44. If 2nC3 : nC2 = 44:3, then for which of the following 53. On the occasion of Deepawali festival each student of
values of r, the value of nCr will be 15 a class sends greeting cards to the others. If there are
;fn 2nC3 : nC2 = 44:3 gS] rks fuEufyf•r esa ls r ds fdl eku ds 20 students in the class, then the total number of
greeting cards exchanged by the students is
fy, nCr dk eku 15 gksxk
nhikoyh ds volj ij ,d d{kk dk çR;sd Nk=k nwljs Nk=kksa dks xzhf
(a) r = 3 (b) r = 4 (c) r = 6 (d) r = 5
dkMZ Hkstrk gSA ;fn20 d{kk
Nk=k
esagSa] rks Nk=kksa }kjk vknku&çnku
45. If nCr–1 = 36, nCr = 84 and nCr+1 = 126, then the value of
x, xzhfVax dkMks± dh dqy la[;k D;k gS\
r is

(a) 1 (b) 2

b y :
;fn nCr–1 = 36, nCr = 84 vkSjnCr+1 = 126, rksr dk eku gS
(c) 3 (d) N.O.T
(a) 20C2
54. If 10Cr = 10
(b) 2.20C2 (c) 2.20P2
Cr+2, then 5Cr equals
;fn 10Cr = 10Cr+2, rks5Cr cjkcj gS
(d) N.O.T

s
n n n
46. Cr + 2 Cr–1 + Cr–2 =
(a) 120 (b) 10 (c) 360 (d) 5

c
(a) n+1Cr (b) n+1Cr+1 (c) n+2
Cr (d) n+2
Cr+1
55. If nCr = 84, nCr–1 = 36 and nCr+1 = 126, then n equals

i r
n
47. Cr + nCr–1 is equal to

t
;fn nCr = 84, nCr–1 = 36 vkSjnCr+1 = 126, rksn cjkcj gS

i
cjkcj gS

a
n
Cr + nCr–1
(a) 8 (b) 9 (c) 10 (d) 5

S
(a) n+1Cr (b) nCr+1 (c) n+1Cr+1 (d) n–1
Cr–1 n n n+1
56. If C3 + C4 > C3, then

m
48. If Cr = Cr+2, then the value of rC2 is
8 8
;fn nC3 + nC4 > n+1C3, rks

49. If
(a) 8
20

th e s
(b) 3

i a
;fn 8Cr = 8Cr+2, rksrC2 dk eku gS

l (c) 5
Cn+2 = nC16, then the value of n is
(d) 2
(a) n > 6 (b) n > 7
57. If n+1C3 = 2nC2, then n =
;fn rksn =
(c) n < 6 (d) N.O.T

a a
n+1
C3 = 2nC2,
;fn 20Cn+2 = nC16, rksn dk eku gS (a) 3 (b) 4 (c) 5 (d) 6

M raj B
(a) 16C3 (b) 30C16 (c) 15C10 (d) 15C15 58. The least value of natural number n satisfying
50. Everybody in a room shakes hand with everybody else. C(n, 5) + C(n, 6) > C(n + 1, 5) is
The total number of hand shakes is 66. The total
C(n, 5) + C(n, 6) > C(n + 1, 5) dks larq"V djus okyh çkÑfrd
number of persons in the room is
la[;k n dk U;wure eku D;k gS

e
,d dejs esa lHkh yksx ,d&nwljs ls gkFk feykrs gSaA gkFk feykus okyksa (b) 10
(a) 11 (c) 12 (d) 13

e
dh dqy la[;k 66 gSA dejs esa dqy O;fDr;ksa dh la[;k gS%59. If n and r are two positive integers such that n  r,
(a) 11 (b) 12 (c) 13 (d) 14 then nCr–1 + nCr =

N
51. In a football championship, there were played 153
matches. Every team played one match with each
other. The number of teams participating in the
championship is
,d iQqVckWy pkSafi;uf'ki
153 eSp
(a) nCn–r
;fn n vkSjr nks /ukRed iw.kkZad bl çdkjngS
n

(b) nCr
60. nC + n–1C +.....+ rCr =
esa •sys x,A çR;sd Vhe us ,d&nwljs r n+1 r
(a) Cr
ds lkFk ,d eSp •sykA pkSafi;uf'ki esa Hkkx ysus okyh Vheksa dh la[;k gS%
Cr–1 + nCr =

(b) n+1Cr+1
(c) n–1Cr

(c) n+2Cr
(d) n+1Cr

(d) 2n
fdrks
 r,

61. How many words can be formed by taking 3

(a) 17 (b) 18 (c) 9 (d) 13 consonants and 2 vowels out of 5 consonants and 4
52. In an examination there are three multiple choice vowels
questions and each question has 4 choices. Number 5 O;atuksa vkSj
4 Lojksa esa 3 O;atu
ls vkSj2 Loj ysdj fdrus 'kCn
of ways in which a student can fail to get all answers
cuk, tk ldrs gSa\
correct, is
,d ijh{kk esa rhu cgqfodYih; ç'u gSa vkSj çR;sd
4 fodYi
ç'u esa (a) 5C3 × 4C2 (b)
5
C3  4C2
gSaA ,d Nk=k fdrus rjhdksa ls lHkh mÙkj lgh djus esa vliQy gks ldrk gS 5

(a) 11 (b) 12 (c) 27 (d) 63 (c) 5C3 × 4C3 (d) (5C3 × 4C2)(5)!
62. Out of 10 white, 9 black and 7 red balls, the number 12 fjfDr;ksa dks Hkjus 25
ds fy,
mEehnokj gSa] ftuesa ls ik¡p vuqlwfpr
of ways in which selection of one or more balls can be tkfr ls gSaA ;fn
3 fjfDr;k¡ vuqlwfpr tkfr ds mEehnokjksa ds fy,
made, is
vkjf{kr gSa vkSj 'ks"k lHkh ds fy, •qyh gSa] rks p;u ds fdrus rjhds
10 liQsn]9 dkyh vkSj
7 yky xsanksa esa ls ,d ;k vf/d xsanksa dk p;u
ldrs gSa
djus ds rjhdksa dh la[;k gS (a) 5C3 × 22C9 (b) 22C9 – 5C3
(a) 881 (b) 891 (c) 879 (d) 892 22 5
(c) C3 + C3 (d) N.O.T

:
63. In a touring cricket team there are 16 players in all
69. A total number of words which can be formed out of
including 5 bowlers and 2 wicket-keepers. How many

y
the letters a, b, c, d, e, f taken 3 together such that
teams of 11 players from these, can be chosen, so as
each word contains at least one vowel, is

b
to include three bowlers and one wicket-keeper
v{kjksa
a, b, c, d, e, f esa3 dks ,d lkFk feykdj cuk, tk ldus okys
,d Hkze.k'khy fØdsV Vhe5 xsanckt
esa vkSj2 fodsVdhij lfgr dqy

s
'kCnksa dh dqy la[;k bl çdkj gS fd çR;sd 'kCn esa de ls de ,d
16 f•ykM+h gSaA buesa
11 f•ykfM+;ksa
ls dh fdruh Vhesa pquh tk ldrh

c
Loj gks]

i
gSa] ftuesa rhu xsanckt vkSj ,d fodsVdhij 'kkfey gksa\

t r
(a) 72 (b) 48 (c) 96 (d) N.O.T

i
(a) 650 (b) 720 (c) 750 (d) 800

a
70. In how many ways can 5 red and 4 white balls be
64. Out of 6 books, in how many ways can a set of one or

S
drawn from a bag containing 10 red and 8 white balls
more books be chosen

m
10 yky vkSj8 liQsn xsanksa okys cSx
5 yky
esavkSj
ls 4 liQsn xsansa
6 iqLrdksa esa ls ,d ;k vf/d iqLrdksa ds lewg dks fdrus rjhdksa ls

(a) 64

th e
pquk tk ldrk gS\
(b) 63

i s la(c) 62 (d) 65
65. Choose the correct number of ways in which 15
fdrus rjhdksa ls fudkyh tk ldrh gSa\
(a) 8C5 ×
(c) 18C9
10
C4 (b) 10C5 × 8C4
(d) N.O.T

(a)
a
M raj B
number of books

52!
a
different books can be divided into five heaps of equal

mu rjhdksa dh lgh la[;k pqusa15

(b)
52!
ftuls
vyx&vyx iqLrdksa dks

(c)
52!
14
71.

cjkcj la[;k esa iqLrdksa ds ikap <sjksa esa foHkkftr fd;k tk lds C4  

(d) N.O.T
4

j1
14

18– j

(a) 18C3
4
C4   18– j C3 is equal to
j1

C3 cjkcj gS

(b) 18C4 (c) 14C7 (d) N.O.T

e
4 2 4
13! 13! 4! 12!  4! 72. The number of ways in which four letters of the word

e
66. How many words of 4 consonants and 3 vowels can 'MATHEMATICS' can be arranged is given by
be formed from 6 consonants and 5 vowels 'MATHEMATICS' 'kCn ds pkj v{kjksa dks O;ofLFkr djus ds rjhdksa

N
6 O;atu vkSj

(a) 75000
5 Lojksa4lsO;atu vkSj
tk ldrs gSa\
3 Lojksa okys fdrus 'kCn cuk,

(b) 756000 (c) 75600 (d) N.O.T

67. In how many ways can 6 persons be selected from 4
officers and 8 constables, if at least one officer is to
be included
dh la[;k bl çdkj nh xbZ gS
(a) 136 (b) 192 (c) 1680 (d) 2454
73. The number of ways in which a committee of 6
members can be formed from 8 gentlemen and 4 ladies
so that the committee contains at least 3 ladies is
8 lTtuksa vkSj
4 efgykvksa6lslnL;ksa dh ,d lfefr cukus ds rjhdksa
4 vf/dkfj;ksa vkSj8 dkaLVscyksa6esa O;fDr;ksa
ls dk p;u fdrus dh la[;k fdruh gS] rkfd lfefr esa de ls de3 efgyk,a gksa\
rjhdksa ls fd;k tk ldrk gS] ;fn de ls de ,d vf/dkjh dks 'kkfey (a) 252 (b) 672 (c) 444 (d) 420
fd;k tkuk gS\ n
74. If Cr denotes the number of combinations of n things
(a) 224 (b) 672 (c) 896 (d) N.O.T taken r at a time, then the expression
n
68. To fill 12 vacancies there are 25 candidates of which Cr+1 + nCr–1 + 2 × nCr equals
five are from scheduled caste. If 3 of the vacancies ;fn nCr ,d le; esa r yh xbZn phtksa ds la;kstuksa dh la[;k dks
are reserved for scheduled caste candidates while the n'kkZrk gS] rks O;atd
n
Cr+1 + nCr–1 + 2 × nCr cjkcj gS
rest are open to all, then the number of ways in which n+2 n+2 n+1 n+1
(a) Cr (b) Cr+1 (c) Cr (d) Cr+1
the selection can be made
75. A student is allowed to select at most n books from a 82. The number of triangles that can be formed by
collection of (2n + 1) books. If the total number of ways choosing the vertices from a set of 12 points, seven of
in which he can select one book is 63, then the value which lie on the same straight line, is
of n is 12 fcanqvksa ds lewg ls 'kh"kks± dks pqudj cuk, tk ldus okys f=kHk
,d Nk=k dks
(2n + 1) iqLrdksa ds laxzg esa ls vf/dre
n iqLrdsa pquus la[;k] ftuesa ls lkr ,d gh lh/h js•k ij fLFkr gSa] gS
dh vuqefr gSA ;fn ,d iqLrd pquus ds mlds dqy rjhdksa dh la[;k(a) 185 (b) 175 (c) 115 (d) 105
63 gS] rks
n dk eku D;k gS\

:
83. There are 16 points in a plane out of which 6 are
(a) 2 (b) 3 (c) 4 (d) N.O.T collinear, then how many lines can be drawn by joining

76. The value of

n–1 n

s
n–1

b y
r 0
n
n
Cr
Cr  nCr 1
is
these points
,d lery esa 16 fcanq gSa ftuesa6 lajs•
ls gSa] rks bu fcanqvksa dks
feykdj fdruh js•k,¡ •haph tk ldrh gSa\

c
Cr
 dk eku gS (a) 106 (b) 105 (c) 60 (d) 55

i
n
Cr  nCr 1

r
r 0 84. There are 16 points in a plane, no three of which are

(a) n + 1

at (b)

S i n
2
(c) n + 2 (d) N.O.T
in a straight line except 8 which are all in a straight
line. The number of triangles that can be formed by
joining them equals

m ,d lery esa 16 fcanq gSa] ftuesa

8 fcanqvksa
ls dks NksM+dj dksbZ Hkh
6

a
50
77. The value of C 4   56–r C3 is

50

th e 6
C4   56–r C3

i s l dk eku gS
r 1 fcanq lh/h js•k esa ugha gSaA bUgsa feykdj cuus okys f=kHkqtk
cjkcj gS
(a) 504 (b) 552 (c) 560 (d) 1120

a a
r 1

(a) 56C3 (b) 56C4 (c) 55C4 (d) 55C3 85. A polygon has 35 diagonals, then the number of its

M raj B
sides is
78. A student is to answer 10 out of 13 questions in an
examination such that he must choose at least 4 from ,d cgqHkqt esa
35 fod.kZ gSa] rks bldh Hkqtkvksa dh la[;k gS
the first five questions. The number of choices (a) 8 (b) 9 (c) 10 (d) 11
available to him is 86. If nPr = 840, nCr = 35, then n is equal to
,d Nk=k dks ,d ijh{kk13 esaesa ls10 ç'uksa ds mÙkj bl çdkj nsus ;fn nP = 840, nC = 35, rksn cjkcj gS

e
gSa fd mls igys ik¡p ç'uksa esa ls de ls de 4 ç'u pquus gSaA mlds(a)
ikl1
r r

e
(b) 3 (c) 5 (d) 7
miyC/ fodYiksa dh la[;k gS 87. The number of divisors of 9600 including 1 and 9600

N
(a) 140 (b) 196 (c) 280 (d) 265 are
79. The number of diagonals in a octagon will be 1 vkSj9600 lfgr 9600 ds Hkktdksa dh la[;k gS
,d v"VHkqt esa fod.kks± dh la[;k gksxh (a) 60 (b) 58 (c) 48 (d) 46
(a) 28 (b) 20 (c) 10 (d) 16
80. If a polygon has 44 diagonals, then the number of its 10   20   m
p 
88. The sum
 
 ,  where    0 if p  q  , is
i 0  i   m – i 
sides are  q  
;fn fdlh cgqHkqt44
esafod.kZ gSa] rks mldh Hkqtkvksa dh la[;k D;k gS\
maximum when m is
(a) 7 (b) 11 (c) 8 (d) N.O.T m
10   20   p 
81. The number of diagonals in a polygon of m sides is ;ksx    ,  where    0 if p  q  , vf/dre gksrk
i 0  i  m – i   q
  
m Hkqtkvksa okys cgqHkqt esa fod.kks± dh la[;k gS
gS] tc m gksrk gS
1 1 (a) 5 (b) 15 (C) 10 (d) 20
(a) m(m – 5) (b) m(m – 1)
2! 2!
89. If 56Pr+6 : 54
Pr+3 = 30800:1, then r =

(c)
1
m(m – 3) (d)
1
m(m – 2) ;fn 56Pr+6 : 54Pr+3 = 30800:1, rksr =
2! 2! (a) 31 (b) 41 (c) 51 (d) N.O.T
90. Ten different letters of an alphabet are given. Words 94. In how many ways can a committee be formed of 5
with five letters are formed from these given letters. members from 6 men and 4 women if the committee
Then the number of words which have at least one has at least one woman
letter repeated is 6 iq#"kksa4vkSj
efgykvksa5lslnL;ksa dh ,d lfefr fdrus rjhdksa ls
,d o.kZekyk ds nl vyx&vyx v{kj fn, x, gSaA bu v{kjksa lscukbZ tk ldrh gS] ;fn lfefr esa de ls de ,d efgyk gks\
ik¡p&ik¡p v{kj okys 'kCn cuk, x, gSaA rks mu 'kCnksa dh la[;k
(a)D;k
186gS (b) 246 (c) 252 (d) N.O.T
ftuesa de ls de ,d v{kj nksgjk;k x;k gS\

:
95. How many words can be made from the letters of the
(a) 69760 (b) 30240 (c) 99748 (d) N.O.T word BHARAT in which B and H never come together

b y
91. A five digit number divisible by 3 has to formed using
the numerals 0, 1, 2, 3, 4 and 5 without repetition.
The total number of ways in which this can be done
is

s
BHARAT 'kCn ds v{kjksa ls ,sls fdrus 'kCn cuk, tk ldrs gSa ftuesa
B vkSjH dHkh ,d lkFk u vk,
(a) 360 (b) 300 (c) 240 (d) 120

c
96. If x, y and r are positive integers, then
ls foHkkT; ,d ik¡p vadksa dh la[;k
0, 1, 2, 3, 4 vkSj5 dk

i
3

t r
x
Cr + xCr–1yC1 + xCr–2yC2 +.....+ yCr =
mi;ksx djds fcuk fdlh nksgjko ds cukuh gSA ,slk djus ds dqy rjhds
gSa
(a) 216

a
(b) 240

m S
(c) 600
i (d) 3125
;fn x, y vkSjr /ukRed iw.kkZad gSa] rks
x
Cr + xCr–1yC1 + xCr–2yC2 +.....+ yCr =

a
92. The number of ways in which an examiner can assign x !y !  x  y  ! (c)

e l
x+y xy
30 marks to 8 questions, awarding not less than 2 (a) (b) Cr (d) Cr
r! r!

h s
marks to any question is

a t i
,d ijh{kd }kjk8 ç'uksa ds fy,30 vad fu/kZfjr djus ds rjhdksa 97.

a
la[;k fdruh gS] rFkk fdlh Hkh ç'u ds fy, de ls de
tk ldrs gSa\
2 vad fn,
dh For 2  r  n,  n   2  n    n  is equal to
r  r – 1  r – 2 

M raj B
 n  n   n 
(a) 21C7 (b) 30C16 (c) 21C16 (d) N.O.T 2  r  n ds fy,]    2    cjkcj gS
r  r – 1  r – 2 
93. Five balls of different colours are to be placed in three
boxes of different sizes. Each box can hold all five  n  1  n  1  n  2  n  2
balls. In how many ways can we place the balls so (a)   (b) 2   (c) 2   (d)  

e
 r –1  r  1   r   r 
that no box remains empty

e
vyx&vyx jaxksa dh ik¡p xsanksa dks vyx&vyx vkdkj ds98. rhu fMCcksa
The number of straight lines that can be formed by
joining 20 points no three of which are in the same
esa j•k tkuk gSA çR;sd fMCck ik¡pksa xsanksa dks lek ldrk gSA ge xsanksa dks

N
straight line except 4 of them which are in the same
fdrus rjhdksa ls j• ldrs gSa rkfd dksbZ Hkh fMCck •kyh u jgs\line
(a) 50 (b) 100 (c) 150 (d) 200 20 fcanqvksa dks feykdj cukbZ tk ldus okyh lh/h js•kvksa dh la[;k
ftuesa ls dksbZ Hkh rhu ,d gh lh/h js•k esa ugha4 gSa]
ds tksflok;
,d gh js•k esa gSa
(a) 183 (b) 186 (c) 197 (d) 185
 Answer Key
1. (c) 11. (b) 21. (b) 31. (c) 41. (a) 51. (b) 61. (d) 71. (b) 81. (c) 91. (a)
2. (a) 12. (a) 22. (c) 32. (c) 42. (c) 52. (d) 62. (c) 72. (d) 82. (a) 92. (a)
3. (b) 13. (c) 23. (a) 33. (b) 43. (c) 53. (b) 63. (b) 73. (a) 83. (a) 93. (c)
4. (c) 14. (b) 24. (c) 34. (d) 44. (b) 54. (d) 64. (b) 74. (b) 84. (a) 94. (b)
5.
6.
7.
(d)
(a)
(c)
15.
16.
17.
(b)
(b)
(c)
25.
26.
27.

b y :
(b)
(d)
(a)
35.
36.
37.
(b)
(b)
(a)
45.
46.
47.
(c)
(c)
(a)
55.
56.
57.
(b)
(a)
(c)
65.
66.
67.
(a)
(b)
(c)
75.
76.
77.
(b)
(b)
(b)
85.
86.
87.
(c)
(d)
(c)
95.
96.
97.
98.
(c)
(c)
(d)
(d)

s
8. (a) 18. (b) 28. (a) 38. (b) 48. (b) 58. (a) 68. (a) 78. (a) 88. (b)

c
9. (d) 19. (b) 29. (c) 39. (d) 49. (d) 59. (d) 69. (c) 79. (b) 89. (b)

i
10. (a) 20. (b) 30. (c) 40. (c) 50. (b) 60. (b) 70. (b) 80. (b) 90. (a)

at S ir
h e m s la
a t ai
M raj B
N e e