Chapter-6

PERMUTATION & COMBINATION

PRACTICE SHEET
1. How many 3digit numbers, each less than 600, (a) 144 (b) 360
can be formed from {1, 2, 3, 4, 7, 9} if repetition of (c) 576 (d) 720
digits is allowed?
(a) 216 (b) 180
12. How many words, with or without meaning can be
formed by using all the letters of the word
(c) 144 (d) 120
‘MACHINE’ so that the vowels occurs only the odd
2. There are four chairs with two chairs in each row. positions?
In how many ways can four persons be seated on (a) 1440 (b) 720
the chairs, so that no chair remains unoccupied? (c) 640 (d) 576
(a) 6 (b) 12
(c) 24 (d) 48 13. From 7 men and 4 women a committee of 6 is to
be formed such that the committee contains at
3. In how many ways can the letters of the world least two women. What is the number of ways to
CORPORATION be arranged so that vowels always do this?
occupy even places? (a) 210 (b) 371
(a) 120 (b) 2700 (c) 462 (d) 5544
(c) 720 (d) 7200
14. If P(32,6) = k C (32, 6), then what is the value of k?
4. In fall permutations of the letters of the ‘LAGAN’ (a) 6 (b) 32
are arranged as in dictionary, then what is the (c) 120 (d) 720
rank of ‘NAAGL’?
(a) 48th Word (b) 49th Word
15. What is the smallest natural number n such that
(c) 50th Word (d) 51st Word n! is divisible by 990?
(a) 9 (b) 11
5. If a secretary and a joint secretary are to be (c) 33 (d) 99
selected from a committee of 11 members, then in
how many ways can they be selected?
16. What is the value of r, if P (5, r) = P (6, r 1)?
(a) 110 (b) 55 (a) 9 (b) 5
(c) 4 (d) 2
(c) 22 (d) 11
6. In how many ways can 7 persons stand in the 17. What is the number of words formed from the
letters of the word ‘JOKE’ so that the vowels and
form of a ring?
(a) P(7, 2) (b) 7! consonants alternate?
(a) 4 (b) 8
(c) 6! (d) 7!/2
(c) 12 (d) None of these
7. On a railway route there are 20 stations. What is 18. If C (n, 12) = C (n, 8), then what is the value of
the number of different tickets required in order C(22, n)?
that it may be possible to travel from every station (a) 131 (b) 231
to every other station? (c) 256 (d) 292
(a) 40 (b) 380
(c) 400 (d) 420 19. In a football championship 153 matches were
played. Every team played one match with each
8. What is the number of five  digit numbers formed other team. How many teams participated in the
with 0, 1, 2, 3, 4 without any repetition of digits? championship?
(a) 24 (b) 48 (a) 21 (b) 18
(c) 96 (d) 120 (c) 17 (d) 15
9. A group consists of 5 men and 5 women. If the 20. How many times does the digit 3 appear while
number of different fiveperson committees writing the integers from1 to 1000?
containing k men and (5k) women is 100, what is (a) 269 (b) 308
the value of k? (c) 300 (d) None of these
(a) 2 only (b) 3 only
(c) 2 or 3 (d) 4 21. In how many ways can a committee consisting of 3
men and 2 women be formed from 7 men and 5
10. If 7 points out of 12 are in the same straight line, women?
then what is the number of triangles formed? (a) 45 (b) 350
(a) 84 (b) 175 (c) 700 (d) 4200
(c) 185 (d) 201
22. What is the number of signals that can be sent by
11. In how many ways can 3 books on Hindi and 3 6 flags of different colours taking one or more at a
books on English be arranged in a row on a shelf, time?
so that not all the Hindi Books are together? (a) 21 (b) 6

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 (c) 720 (d) 1956 (c) 6930 (d) 495
23. What is the number of words that can be formed 25. What is the number of three-digit odd numbers
from the letters of the word ‘UNIVERSAL’, the formed by using the digits 1, 2, 3, 4, 5, 6 if
vowels remaining always together? repetition of digits is allowed? :
(a) 720 (b) 1440 (a) 60 (b) 108
(c) 17280 (d) 21540 (c) 120 (d) 216
24. A team of 8 players is to be chosen from a group of 26. What is the number of ways of arranging the
12 players. Out of the eight players one is to be letters of the word ‘BANANA’ so that no two N’s
elected as captain and another vice-captain. In appear together?
how many ways can this be done? (a) 40 (b) 60
(a) 27720 (b) 13860 (c) 80 (d) 100

ANSWER KEYS
1. c 2. c 3. d 4. b 5. b 6. b 7. b 8. c 9. c 10. c
11. c 12. d 13. b 14. d 15. b 16. c 17. b 18. b 19. b 20. c
21. b 22. d 23. c 24. a 25. b 26. a

Solutions
Sol.1. (c) dictionary order and arranging A, A, L,
 5 
2
There digit number less than 600 will 4!     100
have first element 100, and last element N in different ways, there are = 12  k!5  k ! 
599. First place will not have digit more
2!
than 6, hence, 7 and 9 cannot be taken: words. Next the 37th word starts with L
that comes next in dictionary order 5!
So first digit can be selected in 4 ways.   10
there are 12 words starting with L. This k!5  k !
Second digit can be selected in 6 ways
and since repetitions of digits are accounts up to the 48 words. The 49th This is true of k = 2 or 3.
allowed, third digit can also be selected word is ‘NAAGL’ Sol.10. (c)
in 6 ways: So, number of ways are 4 × 6 Sol.5. (b) Number of triangles formed from 12
× 6 = 144. Selection of 2 members out of 11 has point = 12 C3
11
Sol.2. (c) C2 number of ways 7
First chair can be occupied in 4 ways Since 7 pars are collinear, then C3
11
and second chair can be occupied in 3 C2  55 triangle will not be formed so.
ways, third chair can be occupied in 2 Sol.6. (c) =12C3–7C3
ways and last chair can be occupied in Number of ways in which 7 person can
one ways only. So total number of ways stand in the form of ring = (7 1)! = 6! 12! 7! 12.11.10 7.6.5
= 4 × 3 × 2 × 1 = 24 =   
Sol.7. (b) 3!9! 3!4! 3.2.1 3.2.1
Sol.3. (d) From each railway station, there are 19
CORPORATION is 11 letter word. different tickets to be issued. There are
It has 5 vowels (O, O, O, A, I) and 6 =220–35=185
20 railway station.
consonants (C, R, P, R, T, N) So, total number of tickets = 20× 19 =
In 11 letters, there are 5 even places 380 Sol.11. (c)
(2nd, 4th, 6th, 8th and 10th positions) Sol.8. (c) Total number of arrangement = 6! = 720
5! To make a 5 digit number, 0 cannot Total number of arrangement while all
5 vowels can take 5 even places in come in the beginning. So, it can be the
3! Hindi books are together = 4!×3!= 24 × 6
filled in 4 ways. Rest of the places can
ways = 144
be filled in 4! Ways. So total number of
(∵ Since O is repeated thrice)  The number of ways, in which books
digit formed = 4 × 4! = 4 × 24 = 96
Similarly, 6 consonants can take 6 odd are arranged, while all the Hindi books
Sol.9. (c)
6! K men selected out of 5 and 5  k
are not together
places in ways = 720  144 = 576
2! women out of 5. These are 5 Ck and
(∵ R is repeated twice)
Sol.12. (d)
5 There are three vowels and they have
5 k C
5! 6! four odd places to arrange. Other letters
 Total number of ways =  = 20 According to problem:
3 ! 2! 5C1C  5C5–1C = 100
are four and has four places to arrange.
× 360  The number of words = 4 P3  4!
= 7200 4!
5! 5!   4!  576
Sol.4. (b)    100
Starting with the letter A and arranging
k!5  k ! 5  k !5! (4  3)!
the other four letters, there are 24 Sol.13. (b)
words. There are the first 24 words. The required number of ways
Then starting with G that comes next in =11C6–(7C6  4C0 + 7C5  4C1)

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 11 10  9  8  7  76  Sol.18. (b) Required number of ways
=  7   4 Given C (n, 12) = C (n, 8)
5 4  3 2  2  6 P16 P2 6 P3 6 P4 6 P5 6 P6
nC12 = nC8 = 6 + 30 + 120 + 360 + 720 + 720 =1956
n! n! Sol.23. (c)
=462 – (7 + 84) = 371  
n  12!12! n  8!8! Consider the word UNIVERSAL
1 Total no. of vowels= U,I,E,A=4
Sol.14. (d) 
33 32 n  12!12 1110  9  8! Let us consider these as a single letter
Since P6  k C6 UNIVERSAL
1 Then, total No. of letters =6
32! 32! 
  k. n  8n  9n  10n  11n  12!8! Then, number of ways to arrange
(32  6)! 6!(32  6)! them=6!=720.
1 1
 k  6!  720   But vowels can also arranged in 4! Or
12  11 10  9 n  8n  9 n  10n  11
Sol.15. (b) 24 ways.
(n–8) (n–9) (n-10) (n–11) Hence, total number of ways=720 X 24
Consider option ‘a’
=12  11  10  9 =17280
Let us take n = 9
Since, 9! = n–8=12, n–9=11, n–10=10 and n–11=9 Sol.24. (a)
9×8×7×6×5×4×3×2×1=362880 n=20 Total no. of players=12
Which is not divisible by 990. C(22, n) = 22C20 No. of chose players=8
Now assume, n = 11 22! 22 21 Number of ways to choose 8 players
=   231 from 12
Since 11! = 39916800 2!20! 2
Which is divisible by 990. players
Sol.19. (b)
Thus, required smallest natural number Let total no. of team participated in a
12! 12 1110  9  8!
11
12 C8    495
championship be n. Since every team 8!4! 8!4!
Sol.16. (c) played one match with each other team. Since, out of the 8 players 1 is to be
Given P (5, r) = P (6, r 1) n! elected as
5Pr = 6Pr–1 n C2=153   153 captain and another vice-captain
2! n  2!
therefore number of ways to choose a
n n  1n  2! n n  1 captain

5! 6!   153   153

5  r ! 6  r  1 ! 2! n  2! 2 and a vice-captain.
n(n–1)=306 8 C1 7C1  8  7  56  n
C1  n 
5! 6! n2 – n – 306 = 0 Here, required number of ways =
 
5  r ! 7  r  ! n(n–18) + 17 (n – 18) = 0 495×56 = 27720
n = 18, – 17 Sol.25. (b)
N cannot be negative Total No. of digits=6
5! 6  5! To form a odd numbers we have only 3
  n  –17
7  r ! 7  r  6  r  5  r  ! choice
 n =18
for the unit digits.
Sol.20. (c) Now, Extreme left place can be filled in
(7–r)(6–r) = 6 Before 1000 there are one digit, two 6 ways
digits and three digits numbers. the middle.
42–13r+r2 = 6 Number of ties 3 appear in one digit
Required number of numbers = 6 x 6
number = 20 × 9
x 3 = 108
r2 – 13r + 36 = 0 Number of times 3 appear in two digit
number = 11 × 9 Sol.26. (a)
Number of times 3 appear in three digit Total no. of letters in BANANA = 6
r2 – 9r – 4r + 36 = 0 No. of repeated letter N = 2
numbers = 21
Hence total number of times the digit 3 No. of repeated letter A =2
(r–9) (r–4)=0 appear while writing the integers from 1 Therefore , Number of ways that can be
to 1000 formed
= 180 + 99 + 21= 300 by using the words
r = 4 (r9)
Sol.17. (b) Sol.21. (b) ‘BANANA’ = 6!  6  5  4  3!  60
Total no. of Men=7 3!2! 3! 2!
Total number of letters = 4
Total no. of women=5 Number of ways in which two N comes
No. of vowels = 2
Required number of ways= 7C3X5C2 together= 5! =20
No. of consonants = 2
Possibilities of words formed from the 7! 5! 7  6 5 5 4 3!
letters of word “JOKE” are      Required number of ways
3!4! 2! n ! 3 2 2 = 60 20= 40
JOKE, KOJE, KEJO, JEKO, EJOK,
EKOJ, OKEJ, OJEK 7  5 10  35 10  350
Thus, required number of words = 8 Sol.22. (d)

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 NDA PYQ
1. A, B, C, D and E are coplanar points and three of 10. What is the number of ways that 4 boys and 3
them lie in a straight line. What is the maximum girls can be seated so that boys and girls
number of triangles that can be drawn with these alternate?
points as their vertices? (a) 12 (b) 72
(a) 5 (b) 9 (c) 120 (d) 144
(c) 10 (d) 12 [NDA (I) - 2012]
[NDA (I) - 2011] 11. The number of permutations that can be formed
2. Using the digits 1,2, 3,4 and 5 only once, how from all the letters of the word ‘BASEBALL’ is:
many numbers greater than 41000 can be (a) 540 (b) 1260
formed? (c) 3780 (d) 5040
(a) 41 (b) 48 [NDA (II) - 2012]
(c) 50 (d) 55 12. What is the number of diagonals which can be
[NDA (I) - 2011] drawn by joining the angular points of a polygon
3. What is the value of n, if P(15, n -1): P(16, n - 2) of 100 sides?
=3:4? (a) 4850 (b) 4950
(a) 10 (b) 12 (c) 5000 (d) 10000
(c) 14 (d) 15 [NDA-2012(2)]
[NDA (I) - 2011] 13. In how many ways can the letters of the word
4. In how many ways 6 girls can be seated in two ‘GLOOMY’ be arranged so that the two O’s should
chairs? not be together?
(a) 10 (b) 15 (a) 240 (b) 480
(c) 24 (d) 30 (c) 60 (d) 720
[NDA (I) - 2011] [NDA (I) - 2013]
5. 5 books are to be chosen from a lot of 10 books. If 14. If P (77, 31) =x and C (77, 31) = y, then which one
m is the number of ways of choice when one of the following is correct?
specified book is always included and n is the (a) x = y (b) 2x = y
number of ways of choice when a specified book (c) 77x = 31y (d) x > y
is always excluded, then which one of the [NDA (I) - 2013]
following is correct? 15. A bag contains balls of two colours, 3 black and 3
(a) m > n (b) m = n white. What is the smallest number of balls
(c) m = n - 1 (d) m= n - 2 which must be drawn from the bag, without
[NDA (I) - 2011] looking, so that among these there are two of the
6. What is the total number of combination of n same colour?
different things taken 1, 2, 3,….,n at a time? (a) 2 (b) 3
(a) 2n+1 (b) 22n+1 (c) 4 (d) 5
(c) 2n-1 (d) 2n −1 [NDA-2013(1)]
[NDA (I) - 2011] n
16. What is  Cn, r  equal to?
7. What is the value of 
 ?
n P n, r
r 0
r 1 r! (a) 2n –1 (b) n
(a) 2n-1 (b) 2 n (c) n! (d) 2n
(c) 2n-1 (d) 2 n + 1 [NDA-2013(2)]
[NDA (II) - 2011] 17. If C(28,2r) = C(28, 2r –4), then what is r equal to?
8. There are 4 candidates for the post of a lecturer (a) 7 (b) 8
in Mathematics and one is to be selected by votes (c) 12 (d) 16
of 5 men. What is the number of ways in which [NDA-2013(2)]
the votes can be given? 18. How many different words can be formed by
(a)1048 (b) 1072 taking four letters out of the letters of the word
(c)1024 (d) 625 'AGAIN' if each word has to start with A ?
[NDA (II) - 2011] (a) 6 (b) 12
9. How many diagonals will be there in an n-sided (c) 24 (d) None
regular polygon? [NDA (I) - 2014]
n n  1 n n  3 19. Out of 7 consonants and 4 vowels, words are to
(a) (b)
2 2 be formed by involving 3 consonants and 2

(c) n2–n (d)

n n  1 vowels. The number of such words formed is :
2 (a) 25200 (b) 22500
[NDA (II)-2011] (c) 10080 (d) 5040
[NDA (I) - 2014]

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 For the next three (03) items that follow: 31. A polygon has 44 diagonals. The number of its
Given that C(n,r) : C(n, r+1) = 1:2 and C(n, r+1) : sides is
C(n, r+2) = 2:3. (a) 11 (b) 10
20. What is n equal to? (c) 8 (d) 7
(a) 11 (b) 12 [NDA (II) - 2015]
(c) 13 (d) 14 32. What is the number of four-digit decimal
[NDA-2014(1)] numbers in which no digit is repeated?
21. What is r equal to? (a) 3024 (b) 4536
(a) 2 (b) 3 (c) 5040 (d) None
(c) 4 (d) 5 [NDA (I) - 2016]
[NDA-2014(1)] 33. What is the number of ways in which 3 holiday
22. What is P(n,r) : C(n,r) equal to? travel tickets are to be given to 10 employees of
(a) 6 (b) 24 an organization, if each employee is eligible for
(c) 120 (d) 720 any one or more of the tickets?
[NDA-2014(1)] (a) 60 (b) 120
23. What is the number of ways in which one can (c) 500 (d) 1000
post 5 letters in 7 letters boxes? [NDA (I) - 2016]
(a) 75 (b) 35 34. What is the number of different messages that
(c) 57 (d) 2520 can be represented by three 0’s and two 1’s?
[NDA (II) - 2014] (a) 10 (b) 9
24. What is the number of ways that a cricket team of (c) 8 (d) 7
11 players can be made out of 15 players? [NDA (I) - 2016]
(a) 364 (b) 1001 35. What is the number of odd integers between 1000
(c) 1365 (d) 32760 and 9999 with no digit repeated?
[NDA (II) - 2014] (a) 2100 (b) 2120
25. How many words can be formed using all the (c) 2240 (d) 3331
letters of the word ‘NATION’ so that all the three [NDA (II) - 2016]
vowels should never come together? 36. A five-digit number divisible by 3 is to be formed
(a) 354 (b) 348 using the digits 0, 1, 2, 3 and 4 without repetition
(c) 288 (d) None of these of digits. What is the number of ways this can be
[NDA (I) - 2015] done?
1 (a) 96
26. What is  n  r C n equal to? (b) 48
r 0
(c) 32
(a) n+1C1 (b) n+2Cn (d) No number can be formed
(c) n+3Cn (d) n+2Cn+1 [NDA (II) - 2016]
[NDA-2015(1)] 37. Out of 15 points in a plane, n points are in the
27. The number of ways in which a cricket team of 11 same straight line. 445 triangles can be formed
players be chosen out of a batch of 15 players so by joining these points. What is the value of n?
that the captain of the team is always included, is (a) 3 (b) 4
(a) 165 (b) 364 (c) 5 (d) 6
(c) 1001 (d) 1365 [NDA (II) - 2016]
[NDA (I) - 2015] 5
28. If different words are formed with all the letters of 38. What is 47C4 + 51C3 +
the word ‘AGAIN’ and are arranged alphabetically

j 2
52  j
C3 equal to?

among themselves as in a dictionary, the word at (a) 52C4 (b) 51C5

the 50th place will be (c) 53C4 (d) 52C5
(a) NAAGI (b) NAAIG [NDA (II) - 2016]
(c) IAAGN (d) IAANG 39. The number of different words (eight-letter words)
[NDA (I) - 2015] ending and beginning with a consonant which
29. The number of 3-digit even numbers that can be can be made out of the letters of the word
formed from the digits 0, 1, 2, 3, 4 and 5, 'EQUATION' is
repetition of digits being not allowed, is (a) 5200 (b) 4320
(a) 60 (b) 56 (c) 3000 (d) 2160
(c) 52 (d) 48 [NDA (I) - 2017]
[NDA (II) - 2015] 40. Let A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Then the
30. The number of ways in which 3 holiday tickets number of subsets of A containing two or three
can be given to 20 employees of an organization if elements is:
each employee is eligible for any one or more of (a) 45 (b) 120
the tickets, is (c) 165 (d) 330
(a) 1140 (b) 3420 [NDA-2017(1)]
(c) 6840 (d) 8000 41. Three-digit numbers are formed from the digits 1,
NDA (II) - 2015] 2 and 3 in such a way that the digits are not

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 repeated. What is the sum of such three-digit [NDA (II) - 2018]
numbers? 52. There are 17 cricket players, out of which 5
(a) 1233 (b) 1322 players can bowl .In how many ways can a team
(c) 1323 (d) 1332 of 11 players be selected to as to include 3
[NDA-2017(1)] bowlers?
42. The value of [C(7,0) + C(7,1)] + [C(7,1) + C(7,2)] (a) C(17,11) (b) C(12,8)
+……………..+[C(7,6)+C(7,7)] is (c) C(17,5)×C(5,3) (d) C(5,3) ×C(12,8)
(a) 254 (b) 255 [NDA (II) - 2018]
(c) 256 (d) 257 53. How many three-digit even numbers can be
[NDA-2017(1)] formed using the digits 1, 2, 3, 4 and 5 when
43. A tea party is arranged for 16 people along two repetition of digits is not allowed?
sides of a long table with eight chairs on each (a) 36 (b) 30
side. Four particular men wish to sit on one (c) 24 (d) 12
particular side and two particular men on the [NDA (I) - 2019]
other side. The number of ways they can be 54. From 6 programmers and 4 typists, an office
seated is wants to recruit 5 people. What is the number of
(a) 24 x 8! x 8! (b) (81)3 ways this can be done so as to recruit at least one
(c) 210 x 8! X 8! (d) 16! typist?
[NDA (II) - 2017] (a) 209 (b) 210
44. How many different permutations can be made (c) 246 (d) 242
out of the letters of the word 'PERMUTATION'? [NDA (I) - 2019]
(a) 19958400 (b) 19954800 55. There are 10 points in a plane. No three of these
(c) 19952400 (d) 39916800 points are in a straight line. What is the total
[NDA (II) - 2017] number of straight lines which can be formed by
45. What is the number of triangles that can be joining the points?
formed by choosing the vertices from a set of 12 (a) 90 (b) 45
points in a plane seven of which lie on the same (c) 40 (d) 30
straight line? [NDA (I) - 2019]
(a) 185 (b) 175 56. If c(20, n+2) = C(20, n–2), then what is n equal
(c) 115 (d) 105 to?
[NDA (I) - 2018] (a) 8 (b) 10
46. How many four-digit numbers divisible by 10 can (c) 12 (d) 16
be formed using 1, 5, 0, 6, 7 without repetition of [NDA-2019(1)]
digits? 57. What is C(47,4) + C(51,3) + C(50,3) + C(49,3) +
(a) 24 (b) 36 C(48,3) + C(47,3) equal to?
(c) 44 (d) 64 (a) C( 47,4) (b) C(52,5)
[NDA (I) - 2018] (c) C(52,4) (d) C(47,5)
47. How many numbers between 100 an 1000 can be [NDA – 2019(2)]
formed with the digits 5, 6, 7, 8, 9, if the 58. If n! has 17 zeros, then what is the value of n?
repetition of digits is not allowed? (a) 95
(a) 32 (b) 53 (b) 85
(c) 120 (d) 60 (c) 80
[NDA (I) - 2018] (d) no such value of n exists
48. What is C(n, r) + 2C (n, r–1) + C(n, r–2) equal to? [NDA-2019(2)]
(a) C(n+1, r) (b) C(n–1, r+1) 59. If P(n,r) = 2520 and C(n,r) = 21, then what is the
(c) C(n, r+1) (d) C(n+2, r) value of C( n + 1, r + 1) ?
[NDA-2018(1)] (a) 7 (b) 14
49. Let x be the number of integers lying between (c) 28 (d) 56
2999 and 8001 which have at least two digits [NDA (II) - 2019]
equal. Then x is equal to: 60. What is the number of diagonals of an octagon?
(a) 2480 (b) 2481 (a) 48 (b) 40
(c) 2482 (d) 2483 (c) 28 (d) 20
[NDA-2018(2)] [NDA-2019(2)]
50. What is the sum of all three-digit numbers that 61. If C(20, n + 2) = C(20, n − 2) , then what is n
can be formed using all the digits 3, 4 and 5, equal to ?
when repetition of digits is not allowed? (a) 18 (b) 25
(a) 2664 (b) 3882 (c) 10 (d) 11
(c) 4044 (d) 4444 [NDA 2020]
[NDA (II) - 2018] 62. What is the number of ways in which the letters
51. The total number of 5-digit numbers that can be of the word 'ABLE' can be arranged so that the
composed of distinct digits from 0 to 9 is vowels occupy odd places?
(a) 45360 (b) 30240 (a) 2 (b) 4
(c) 27216 (d) 15120 (c) 6 (d) 8

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 [NDA 2020] (c) 75 (d) 100
63. What is the maximum number of points of [NDA (II) - 2021]
intersection of 5 non-overlapping circles 73. If C(3n, 2n) = C(3n, 2n–7), then what is the value
(a) 10 (b) 15 of C(n, n–5)?
(c) 20 (d) 25 (a) 42 (b) 35
[NDA 2020] (c) 28 (d) 21
64. How many 5-digit prime numbers can be formed [NDA (I) - 2022]
by using the digits 1, 2, 3, 4 5. If the repetition of 74. What is the value of
digits is not allowed? C(51,21)–C(51,22)+C(51,23)–C(51,24) +C(51,25)–
(a) 5 (b) 4 C(51,26)+C(51,27)–C(51,28) +C(51,29)–C(51,30)?
(c) 3 (d) 0 (a) C(51,25) (b) C(51, 27)
[NDA (I) - 2021] (c) C(51,51)–C(51,0) (d) C(51,25)–C(51,27)
65. In how many ways can a team of 5 players be [NDA (I) - 2022]
selected from 8 players so as not to include a 75. How many odd numbers between 300 and 400 are
particular player? there in which none of the digits is repeated?
(a) 42 (b) 35 (a) 32 (b) 36
(c) 21 (d) 20 (c) 40 (d) 45
[NDA (I) - 2021] [NDA (I) - 2022]
66. If an = n (n!), then what is a1 + a2 + a3 + …. + a10 76. Consider the following statements:
(a) 10! – 1 (b) 11! + 1 n!
(c) 10! + 1 (d) 11! – 1 1. is divisible by 6, where n > 3
3!
[NDA (II) 2021]
67. Let S = (2, 3, 4, 5, 6, 7, 9). How many different 3- n!
2.  3 is divisible by 7, where n > 3
digit numbers (with all digits different) from S can 3!
be made which are less than 500? Which of the above statements is/are correct?
(a) 30 (b) 49 (a) 1 only (b) 2 only
(c) 90 (d) 147 (c) Both 1 and 2 (d) Neither 1 nor 2
[NDA (II) - 2021] [NDA (I) - 2022]
68. Consider the digit 3, 5, 7, 9. What is the number 77. In how many ways can a term of 5 players be
of 5-digit numbers formed by these digits in which selected out of 9 players so as to exclude two
each of these four digits appears? particular players?
(a) 240 (b) 180 (a) 14 (b) 21
(c) 120 (d) 60 (c) 35 (d) 42
[NDA (II) - 2021] [NDA (I) - 2022]
69. If C (n, 4),C(n, 5) and C(n, 6) are in AP, then what 78. How many permutations are there of the letters of
is the value of n? the word ‘TIGER’ in which the vowels should not
(a) 7 (b) 8 occupy the even positions?
(c) 9 (d) 10 (a) 72 (b) 36
[NDA (II) - 2021] (c) 18 (d) 12
70. How many 4 letter words (with or without [NDA (I) - 2022]
meaning) containing two vowels can be 79. What is the maximum value of n such that 5 th
constructed using only the letters (without divides (30!+35!), where n is a natural number?
repetition) of the word ‘LUCKNOW’? (a) 4 (b) 6
(a) 240 (b) 200 (c) 7 (d) 8
(c) 150 (d) 120 [NDA (I) - 2022]
[NDA (II) - 2021] 80. What is the value of 2 (21) + 3 (3  2  1) + 4 (4 
71. Suppose 20 distinct points are placed randomly on 3  2  1) + 5 (5  4  3  2  1) + …. …. …. + 9
a circle. Which of the following statements is/are (9  8  7  6  5  4  3  2  1) + 2?
correct?
(a) 11! (b) 10!
1.The number of straight lines that can be drawn (c) 10 + 10! (d) 11 + 10!
by joining any two of these points is 380.
[NDA (I) - 2022]
2.The number of triangles that can be drawn by 81. How many four digit natural numbers are there
joining any three of these points is 1140.
such that all of the digits are odd?
Select the correct answer using the code given (a) 625 (b) 400
below: (c) 196 (d) 120
(a)1 only (b) 2 only
[NDA 2022 (II)]
(c)Both 1 and 2 (d) Neither 1 nor 2
82. If different permutations of the letters of the word
[NDA (II) - 2021] ‘MATHEMATICS’ are listed as in a dictionary, how
72. Consider a regular polygon with 10 sides. What is
many words (with or without meaning) are there in
the number of triangles that can be formed by the list before the first word that starts with C?
joining the vertices which have no common side
(a) 302400 (b) 403600
with any of the sides of the polygon? (c) 907200 (d) 1814400
(a) 25 (b) 50
[NDA 2022 (II)]

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 Consider the following for the next three (03) (a) 20% (b) 25%
items that follow: 100 110
Consider the word ‘QUESTION’: (c) % (d) %
83. How many 4-letter words each of two vowels and
3 3

two consonants with or without meaning, can be [NDA – 2023 (1)]

formed? 90. Consider the following statements:
(a) 36 (b) 144 1.(25)! + 1 is divisible by 26
(c) 576 (d) 864 2.(6)! + 1 is divisible by 7
[NDA 2022 (II)] Which of the above statement is/are correct?
84. How many 8-letter words with or without meaning, (a) 1 only (b) 2 only
can be formed such that consonants and vowels (c) both 1 and 2 (d) neither 1 nor 2
occupy alternate positions? [NDA – 2023 (1)]
(a) 288 (b) 576 91. What is the number of 6-digit numbers that can
(c) 1152 (d) 2304 be formed only by using 0, 1, 2, 3, 4 and 5 (each
[NDA 2022 (II)] once); and divisible by 6?
85. How many 8-letter words with or without meaning, (a) 96 (b) 120
can be formed so that all consonants are together? (c) 192 (d) 312
(a) 5760 (b) 2880 [NDA – 2023 (1)]
(c) 1440 (d) 720 Consider the following for the next two (02)
[NDA 2022 (II)] items that follow:
86. Consider the following statements for a fixed Consider the sum S = 0! + 1! + 2! + 3! + 4! + …… +
natural number n: 100!
1.C(n, r) is greatest if n = 2r 92. If the sum S is divided by 8, what is the
2.C(n, r) is greatest if n = 2r – 1 and n = 2r + 1. remainder?
Which of the statements given above is/are (a) 0
correct? (b) 1
(a) 1 only (b) 2 only (c) 2
(c) both 1 and 2 (d) Neither 1 nor 2 (d) cannot be determined
[NDA – 2023 (1)] [NDA – 2023 (1)]
87. M parallel lines cut n parallel lines giving rise to 93. If the sum S is divided by 60, what is the
60 parallelograms. What is the value of (m + n)? remainder?
(a) 6 (b) 7 (a) 1 (b) 3
(c) 8 (d) 9 (c) 17 (d) 34
[NDA – 2023 (1)] [NDA – 2023 (1)]
88. Let x be the number of permutation of the word 94. If 1! + 3! + 5! + 7! + …..+ 199! is divided by 24,
‘PERMUTATIONS’ and y be the number of what is the remainder?
permutations of the word ‘COMBINATIONS’. Which (a) 3 (b) 6
one of the following is correct? (c) 7 (d) 9
(a) x = y (b) y = 2x [NDA-2023 (2)]
(c) x = 4y (d) y = 4x 95. What is the maximum number of points of
[NDA – 2023 (1)] intersection of 10 circles?
89. 5-digit numbers are formed using the digits 0, 1, (a) 45 (b) 60
2, 4, 5 without repetition. What is the percentage (c) 90 (d) 120
of numbers which are greater than 50,000. [NDA-2023 (2)]

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 ANSWER KEY

1. b 2. b 3. c 4. d 5. b 6. d 7. a 8. d 9. b 10. d
11. d 12. a 13. a 14. d 15. a 16. d 17. b 18. c 19. a 20. d

21. c 22 b 23. a 24. c 25. c 26. a 27. c 28. b 29. c 30. d

31. a 32. b 33. d 34. a 35. c 36. d 37. c 38. b 39. b 40. a

41. d 42. c 43. c 44. a 45. a 46. a 47. d 48. d 49. b 50. a

51. c 52. d 53. d 54. c 55. b 56. b 57. c 58. d 59. c 60. d

61. c 62. b 63. c 64. d 65. b 66. d 67. c 68. a 69. a 70. a

71. b 72. b 73. d 74. c 75. a 76. d 77. b 78. b 79. c 80. b

81. a 82. c 83. d 84. c 85. b 86. c 87. d 88. c 89. b 90. b

91. d 92. c 93. d 94. c 95. c

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