Class 11 Mathematics Permutations and Combinations MCQs Set 05

Question. There are 3 routes from Guntur to Madras and 4 routes from Madras to Cochin, in how many different ways a person can travel from Guntur to Cochin via Madras is
(a) 21
(b) 24
(c) 12
(d) 20
Answer: (c) 12

Question. The number of permutations of 8 things taken ‘r’ at a time is 1680. Then r =
(a) 4
(b) 5
(c) 6
(d) 7
Answer: (a) 4

Question. The number of permutations of ‘n’ dissimilar things taken 'r' at a time, in which a particular thing always occur is
(a) \( {}^{(n-1)}P_{(r-1)} \)
(b) \( r \cdot {}^{(n-1)}P_{(r-1)} \)
(c) \( r \cdot {}^{(n-1)}P_{r} \)
(d) \( r! \cdot {}^{(n-1)}P_{(r-1)} \)
Answer: (b) \( r \cdot {}^{(n-1)}P_{(r-1)} \)

Question. \( 1 \cdot {}^{1}P_{1} + 2 \cdot {}^{2}P_{2} + \dots + 6 \cdot {}^{6}P_{6} = \)
(a) 720
(b) 5039
(c) 7!
(d) 8!
Answer: (b) 5039

Question. The remainder obtained when 1!+2!+......+100! is divided by 240 is
(a) 153
(b) 154
(c) 155
(d) 156
Answer: (a) 153

Question. The number of ways in which the candidates \( A_1, A_2, \dots, A_{10} \) can be ranked if \( A_1 \) and \( A_2 \) are next to each other is
(a) 9! 2!
(b) 9!
(c) \( \frac{10!}{2!} \)
(d) \( \frac{9!}{4!} \)
Answer: (a) 9! 2!

Question. The number of ways in which the candidates \( A_1, A_2, \dots, A_{10} \) can be ranked if \( A_1 \) is always above \( A_2 \) is
(a) 9! 2!
(b) 9!
(c) \( \frac{10!}{2} \)
(d) 10!
Answer: (c) \( \frac{10!}{2} \)

Question. The number of ways in which 10 candidates \( A_1, A_2, A_3, A_4, \dots, A_{10} \) can be ranked if \( A_1 \) is just above \( A_2 \) is
(a) 9! 2!
(b) 10!
(c) 10! 2!
(d) 9!
Answer: (d) 9!

Question. The number of words that can be formed from the letters of the word "INTERMEDIATE" in which no two vowels are together is
(a) \( 6! \cdot {}^{7}P_{6} \)
(b) \( \frac{6!}{2!} \cdot \frac{{}^{7}P_{6}}{2! 3!} \)
(c) \( \frac{6!}{2! 3!} \cdot \frac{{}^{7}P_{6}}{2! 3!} \)
(d) \( \frac{7! \cdot {}^{7}P_{6}}{2! 5!} \)
Answer: (b) \( \frac{6!}{2!} \cdot \frac{{}^{7}P_{6}}{2! 3!} \)

Question. The number of ways to rearrange the letters of the word CHEESE is
(a) 119
(b) 240
(c) 720
(d) 6
Answer: (a) 119

Question. If the letters of word ‘VICTORY’ are arranged in the order of a dictionary. Then rank of the word ‘VICTORY’ is
(a) 3731
(b) 3732
(c) 3733
(d) 3720
Answer: (c) 3733

Question. The rank of the word ‘SEASON’ is
(a) 210
(b) 220
(c) 230
(d) 270
Answer: (d) 270

Question. The number of ways in which 5 boys and 3 girls can sit around a table so that all the girls do not come together is
(a) 4020
(b) 4120
(c) 4220
(d) 4320
Answer: (d) 4320

Question. The number of ways that a garland is made with 18 flowers such that the two specified flowers should be side by side in the garland is
(a) 15
(b) 120
(c) 17!
(d) 16!
Answer: (d) 16!

FUNDAMENTAL PRINCIPLE

Question. A student has 5 pants and 8 shirts. The number of ways in which he can wear the dress in different combinations is
(a) \( ^{8}P_{5} \)
(b) \( ^{8}C_{5} \)
(c) \( 8! \times 5! \)
(d) 40
Answer: (d) 40

Question. An automobile dealer provides motor cycles and scooters in three body patterns and 4 different colours each. The number of choices open to a customer is
(a) \( ^{5}C_{3} \)
(b) \( ^{4}C_{3} \)
(c) \( 4 \times 3 \)
(d) \( 4 \times 3 \times 2 \)
Answer: (d) \( 4 \times 3 \times 2 \)

Question. In a class there are 10 boys and 8 girls. The teacher wants to select either a boy or a girl to represent the class in a function. The number of ways the teacher can make this selection.
(a) 18
(b) 80
(c) \( ^{10}P_{8} \)
(d) \( ^{10}C_{8} \)
Answer: (a) 18

PROBLEMS BASED ON \( ^{n}P_{r} \) FORMULA

Question. If \( ^{9}P_{5} + 5 \cdot ^{9}P_{4} = ^{10}P_{r} \) then \( r = \)
(a) 4
(b) 5
(c) 6
(d) 7
Answer: (b) 5

Question. If \( \frac{^{20}P_{r-1}}{a} = \frac{^{20}P_{r}}{3} = \frac{^{20}P_{r+1}}{4} \) then \( a = \)
(a) 9/7
(b) 7/9
(c) 3/4
(d) 4/3
Answer: (a) 9/7

Question. If \( ^{(2n+1)}P_{n-1} : ^{(2n-1)}P_{n} = 3 : 5 \) then \( n = \)
(a) 4
(b) 5
(c) 6
(d) 3
Answer: (a) 4

Question. If \( ^{n}P_{7} = 42 \cdot ^{n}P_{5} \) then \( n = \) [EAM-98]
(a) 5
(b) -1
(c) 7
(d) 12
Answer: (d) 12

Question. If \( ^{15}P_{r} = 32760 \), then \( r = \)
(a) 4
(b) 5
(c) 6
(d) 7
Answer: (a) 4

Question. If \( ^{n}P_{r} = 5040 \) then \( (n, r) = \)
(a) (9, 4)
(b) (10, 4)
(c) (11, 3)
(d) (11, 4)
Answer: (b) (10, 4)

Question. The value of \( \sum_{r=1}^{10} r \cdot P(r, r) \) is
(a) \( P(11, 11) \)
(b) \( P(11, 11) - 1 \)
(c) \( P(11, 11) + 1 \)
(d) \( 12! - 1 \)
Answer: (b) \( P(11, 11) - 1 \)

PROBLEMS BASED ON REMAINDER & OTHER MODELS

Question. The value of \( 1 + 1 \cdot 1! + 2 \cdot 2! + 3 \cdot 3! + \dots + n \cdot n! \) is
(a) \( (n+1)!+1 \)
(b) \( (n-1)!+1 \)
(c) \( (n+1)!-1 \)
(d) \( (n+1)! \)
Answer: (d) \( (n+1)! \)

Question. The last two digits in \( X = \sum_{k=1}^{100} k! \) are
(a) 10
(b) 11
(c) 12
(d) 13
Answer: (d) 13

Question. The remainder obtained when \( 1!+2!+ \dots 49! \) is divided by 20 is
(a) 13
(b) 33
(c) 12
(d) 11
Answer: (a) 13

Question. When \( n! + 1 \) is divided by any natural number between 2 and n then remainder obtained is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (a) 1

Question. If \( \sum_{k=1}^{m} (k^{2} + 1)k! = 1999(2000!) \), then m is
(a) 1999
(b) 2000
(c) 2001
(d) 2002
Answer: (a) 1999

PERMUTATIONS OF DISSIMILAR THINGS

Question. The number of different signals can be given by using any number of flags from 4 flags of different colours is
(a) 24
(b) 256
(c) 64
(d) 60
Answer: (c) 64

Question. Three Men have 4 coats 5 waist Coats, and 6 caps. The number of ways they can wear them is
(a) \( ^{15}P_{3} \)
(b) \( 4^{3} 5^{3} 6^{3} \)
(c) \( ^{4}P_{3} \cdot ^{5}P_{3} \cdot ^{6}P_{3} \)
(d) 180
Answer: (c) \( ^{4}P_{3} \cdot ^{5}P_{3} \cdot ^{6}P_{3} \)

Question. A railway carriage can seat 5 each side. The number of ways a party of 4 girls and 6 boys can seat themselves so that the girls may always have the corner seats is
(a) 17,430
(b) 17,431
(c) 17,280
(d) 17,281
Answer: (c) 17,280

Question. The number of words that can be formed using any number of letters of the word "KANPUR" is
(a) 720
(b) 1956
(c) 360
(d) 370
Answer: (b) 1956

Question. The number of words that can be formed using all the letters of the word "KANPUR" when the vowels are in even places is
(a) 144
(b) 36
(c) 24
(d) 48
Answer: (a) 144

Question. The letters of the word "LOGARITHM" are arranged in all possible ways. The number of arrangements in which the relative positions of the vowels and consonants are not changed is
(a) 4320
(b) 720
(c) 4200
(d) 3420
Answer: (a) 4320

Question. The number of ways one can arrange words with the letters of the word "MADHURI" so that always vowels occupy the beginning, middle and end places is
(a) 7!
(b) \( ^{3}C_{3} \cdot ^{4}P_{3} \)
(c) \( 3 \cdot ^{4}C_{4} \)
(d) \( 3! \cdot 4! \)
Answer: (d) \( 3! \cdot 4! \)

Question. The letters of the word "FLOWER" are taken 4 at a time and arranged in all possible ways. The number of arrangements which begin with 'F' and end with 'R' is
(a) 20
(b) 18
(c) 14
(d) 12
Answer: (d) 12

Question. The number of permutations that can be made from the letters of the word "SUNDAY" without beginning with 'S' or without ending with 'Y' is
(a) 696
(b) 624
(c) 604
(d) 504
Answer: (a) 696

Question. Ten guests are to be seated in a row of which three are ladies. The ladies insist on sitting together while two of the gentlemen refuse to take consecutive seats. In how many ways can the guests be seated?
(a) 2399976
(b) 21844
(c) 630624
(d) 181440
Answer: (d) 181440

Question. The number of ways in which 6 Boys and 5 Girls can sit in a row so that all the girls maybe together is
(a) \( 6! \cdot 5! \)
(b) \( 6! \cdot ^{7}P_{5} \)
(c) \( (6!)^{2} \)
(d) \( 7! \cdot 5! \)
Answer: (d) \( 7! \cdot 5! \)