Class 11 Mathematics Combinations MCQs Set 03

Question. If a polygon of n sides has 275 diagonals, then n = 
(a) 25
(b) 35
(c) 20
(d) 15
Answer: (a) 25

Question. A parallelogram is cut by two sets of m lines parallel to the sides. The number of parallelograms thus formed is
(a) \( \frac{m^2}{4} \)
(b) \( \frac{(m+1)^2}{4} \)
(c) \( \frac{(m+2)^2}{4} \)
(d) \( \frac{(m+1)^2(m+2)^2}{4} \)
Answer: (d) \( \frac{(m+1)^2(m+2)^2}{4} \)

Question. The number of rectangles on a chess board is
(a) 1296
(b) 204
(c) 1292
(d) 200
Answer: (a) 1296

Question. The number of ways of choosing 2 squares \( (1 \times 1) \) from a chess board so that they have exactly one common corner is
(a) 98
(b) 112
(c) 36
(d) 72
Answer: (a) 98

Question. If \( t_n \) denotes the number of triangles formed with n points in a plane no three of which are collinear and if \( t_{n+1} - t_n = 36 \), then \( n = \) 
(a) 7
(b) 8
(c) 9
(d) 10
Answer: (c) 9

Question. A polygon has 54 diagonals. Then the number of its sides is 
(a) 7
(b) 9
(c) 10
(d) 12
Answer: (d) 12

Question. Note the arrangements (1, 1), (1, 2), (1, 3), (2, 3), (3, 3), (3, 4), (4, 4). Have we start from (1, 1), then increase one of the coordinates by 1 and repeat the same until we reach (4, 4). For example (1, 1), (2, 1), (2, 2), (2, 3), (3, 3), (3, 4), (4, 4) is another such arrangements. The number of such arrangements is
(a) \( ^6P_6 \)
(b) \( ^6C_6 \)
(c) \( \frac{6!}{3!3!} \)
(d) 6
Answer: (a) \( ^6P_6 \)

TOTAL NUMBER OF COMBINATIONS

Question. The number of ways of selecting atleast one red ball from a bag containing 4 identical red balls and 5 identical black balls is
(a) 20
(b) 21
(c) 23
(d) 24
Answer: (d) 24

Question. Given 5 different green, 4 different blue, and 3 different red dice. The number of combinations of dice that can be choosen by taking atleast one green and one blue die is
(a) 3680
(b) 3690
(c) 3700
(d) 3720
Answer: (d) 3720

Question. A zoo has 20 zebras, 12 Giraffes, 11 lions and 3 tigers. The number of ways a tourist can visit the animals so that he must see at least one tiger is
(a) \( 21 \times 13 \times 12 \times 3 \)
(b) \( 7 \times 2^{43} \)
(c) \( 7 \times 21 \times 13 \times 12 - 1 \)
(d) \( 6 \times 2^{43} \)
Answer: (b) \( 7 \times 2^{43} \)

Question. At an election a voter may vote for any number of candidates not greater than the number to be choosen. There are 10 candidates and 5 members are to be chosen. The number of ways in which a voter may vote is
(a) 630
(b) 632
(c) 637
(d) 640
Answer: (c) 637

DISTRIBUTION OF DISSIMILAR THINGS INTO GROUPS

Question. At an election 3 wards of a town are canvassed by 4, 5 and 3 men respectively. If there are 20 men volunteering the number of ways they can be alloted to the different wards is
(a) \( \frac{20!}{3! 4! 5!} \)
(b) \( \frac{12!}{3! 4! 5!} \)
(c) \( \frac{20!}{3! 4! 5! 8!} \)
(d) \( \frac{12!}{3! 4! 5! 8!} \)
Answer: (c) \( \frac{20!}{3! 4! 5! 8!} \)

Question. The number of ways in which 12 balls can be divided between two friends, one receiving 8 and the other 4, is
(a) \( \frac{12!}{8! 4!} \)
(b) \( \frac{12! \cdot 2!}{8! 4!} \)
(c) \( \frac{12!}{8! 4! 2!} \)
(d) \( \frac{12!}{4!} \)
Answer: (b) \( \frac{12! \cdot 2!}{8! 4!} \)

Question. The number of ways of dividing 15 books into 3 groups of 3,4,8 books respectively is
(a) \( \frac{15!}{2! 3! 4! 8!} \)
(b) \( \frac{15!}{(3!)^2 4! 8!} \)
(c) \( \frac{15!}{4! 8!} \)
(d) \( ^{15}C_3 \cdot ^{12}C_4 \cdot ^8C_8 \)
Answer: (d) \( ^{15}C_3 \cdot ^{12}C_4 \cdot ^8C_8 \)

DISTRIBUTION OF SIMILAR THINGS INTO GROUPS

Question. The number of ways in which 13 gold coins can be distributed among three persons such that each one gets at least two gold coins is
(a) 36
(b) 24
(c) 12
(d) 6
Answer: (a) 36

NUMBER OF DIVISORS

Question. The number of positive divisors of \( 2^5 3^6 7^3 \) is
(a) 14
(b) 167
(c) 168
(d) 210
Answer: (c) 168

Question. Number of even divisors of 1600 is
(a) 21
(b) 18
(c) 3
(d) 6
Answer: (b) 18

APPLICATIONS OF ONTO FUNCTION

Question. The no. of 5 digit numbers that can be made using the digits 1 and 2 and in which atleast one digit is different is
(a) 31
(b) 32
(c) 30
(d) 29
Answer: (c) 30

SELECTION OF DISSIMILAR THINGS

Question. A father with 8 children takes them 3 at a time to the zoological garden, as often as he can without taking the same 3 children together more than once. Then the number of times a particular child will not go to the Zoological garden is
(a) 56
(b) 21
(c) 18
(d) 35
Answer: (d) 35

Question. A party of 9 persons are to travel in two vehicles, one of which will not hold more than 7 and the other not more than 4. The number of ways the party can travel is
(a) 120
(b) 220
(c) 236
(d) 246
Answer: (d) 246

Question. Out of 9 boys the number to be taken to form a group, so that the number of different groups may be greatest is
(a) 4
(b) 5
(c) 4 or 5
(d) 6
Answer: (c) 4 or 5

Question. A train going from Vijayawada to Hyderabad stops at nine intermediate stations. Six persons enter the train during the journey with six different tickets of the same class. The no. of different tickets they may have
(a) \( ^{11}C_6 \)
(b) \( ^{45}C_6 \)
(c) \( ^9C_6 \)
(d) \( ^{10}C_6 \)
Answer: (b) \( ^{45}C_6 \)

COMMITTE

Question. A committee of 5 men and 3 women is to be formed out of 7 men and 6 women. If two particular women are not to be together in the committee, the number of committees formed is
(a) 420
(b) 5040
(c) 336
(d) 216
Answer: (c) 336

Question. A reserve of 12 railway station masters is to be divided into two groups of 6 each one for day duty and the other for night duty. The number of ways in which this can be done if two specified persons A, B should not be included in the same group is
(a) 500
(b) 504
(c) 508
(d) 512
Answer: (b) 504

Question. A committee of 6 is chosen from 10 men and 7 women so as to contain atleast 3 men and 2 women. If 2 particular women refuse to serve on the same committee, the number of ways of forming the committee is
(a) 7800
(b) 8610
(c) 810
(d) 8000
Answer: (a) 7800

Question. The crew of an 8 oar boat is to be choosen from 12 Men, of whom 3 can row on stroke side only. If selected the number of ways the crew can be arranged is
(a) \( ^9C_4 \cdot ^8C_4 \)
(b) \( 14 \times 9! \)
(c) \( ^{12}C_8 \cdot ^9C_4 \cdot 4! \cdot 4! \)
(d) \( ^{12}C_8 \cdot ^8C_4 \cdot 4! \cdot 4! \)
Answer: (b) \( 14 \times 9! \)

Question. A car will hold 2 persons in the front seat and 1 in the rear seat. If among 6 persons only 2 can drive, the number of ways, in which the car can be filled is
(a) 10
(b) 18
(c) 20
(d) 40
Answer: (d) 40

GEOMETRICAL APPLICATIONS

Question. There are ‘m’ points on a straight line AB and ‘n’ points on another straight line AC in which A is not included. By joining these points triangles are constructed. i) When A is not included. ii) When A is included, the ratio of number of triangles in both cases is
(a) \( \frac{m+n-2}{m+n} \)
(b) \( \frac{m+n-2}{2} \)
(c) \( \frac{m+n-2}{m+n+2} \)
(d) \( \frac{m+n+2}{m+n-2} \)
Answer: (a) \( \frac{m+n-2}{m+n} \)

Question. In a polygon no three diagonals are concurrent. If the total number of points of intersection of diagonals interior to the polygon is 35 and the number of diagonals is 'x', number of sides is 'y' then (y, x) =
(a) (5, 5)
(b) (6, 9)
(c) (5, 20)
(d) (7, 14)
Answer: (d) (7, 14)

Question. Let \( T_n \) denote the number of triangles which can be formed using the vertices of a regular polygon of n sides. If \( T_{n+1} - T_n = 10 \), then the value of n is [AIEEE-2013]
(a) 5
(b) 10
(c) 8
(d) 7
Answer: (a) 5

Question. There are three coplanar lines, if any m points taken on each of the lines, the maximum number of triangle with vertices at these points is
(a) \( m^2(4m-3) \)
(b) \( 3m^2(m-3)+1 \)
(c) \( 3m^2(m-3) \)
(d) \( m(4m-3) \)
Answer: (a) \( m^2(4m-3) \)

Question. ABCD is a convex quadrilateral 3, 4, 5 and 6 points are marked on the sides AB, BC, CD and DA respectively, the number of triangles with vertices on different sides is
(a) 270
(b) 220
(c) 282
(d) 342
Answer: (d) 342

Question. The greatest number of points of intersection of 8 lines and 4 circles is
(a) 64
(b) 92
(c) 104
(d) 96
Answer: (c) 104

DISTRIBUTION OF DISSIMILAR THINGS

Question. A guard of 15 men is formed form a group of n soldiers. The number of times two particular soldiers will guards
(a) \( \frac{n!}{13!2!} \)
(b) \( \frac{(n-2)!}{13!} \)
(c) \( \frac{(n-2)!}{13!(n-15)!} \)
(d) \( \frac{(n-2)!}{13!12!} \)
Answer: (c) \( \frac{(n-2)!}{13!(n-15)!} \)

Question. 15 Passengers are to travel by a double decker bus which can accomodate 5 in upper deck and 10 in lower deck. The number of ways that the passengers are distributed is
(a) 3000
(b) 3003
(c) 3006
(d) 3009
Answer: (b) 3003

DISTRIBUTION OF SIMILAR THINGS

Question. The number of ways of distributing 9 identical balls in 3 distinct boxes so that none of the boxes is empty is
(a) \( ^8C_3 \)
(b) 28
(c) \( 3^8 \)
(d) 5
Answer: (b) 28

Question. Five distinct letters are to be transmitted through a communication channel. A total number of 15 blanks is to be inserted between the the letters with at least three between every two. The number of ways in which this can be done is
(a) 1200
(b) 1800
(c) 2400
(d) 3000
Answer: (c) 2400

Question. The number of nonnegative integer solutions of the equation x + y + z + 5t = 15 is
(a) 196
(b) 224
(c) 312
(d) 364
Answer: (b) 224

Question. The number of integral solutions to the system of equations \( x_1 + x_2 + x_3 + x_4 + x_5 = 20 \) and \( x_1 + x_2 = 15 \) when \( x_k \ge 0 (k=1,2,3,4,5) \) is
(a) 300
(b) 350
(c) 336
(d) 316
Answer: (c) 336

Question. If N is the number of positive integral solutions of \( x_1 x_2 x_3 x_4 = 770 \), then N =
(a) 256
(b) 729
(c) 900
(d) 770
Answer: (a) 256