Class 11 Mathematics Combinations MCQs Set 04

SELECTION OF DISSIMILAR THINGS

Question. A father with 6 children takes 3 at a time to a park without taking the same children. How often each child goes to the park ?
(a) 10
(b) 12
(c) 15
(d) 20
Answer: (a) 10

Question. A person wishes to make up as many different parties of 10 as he can out of 20 friends, each party consisting of the same number. In how many of the parties the same man is found ?
(a) 380
(b) 19
(c) 90378
(d) 92378
Answer: (d) 92378

Question. \( ^{n}C_{r+1} + 2 \cdot ^{n}C_r + ^{n}C_{r-1} = \)
(a) \( ^{n+1}C_r \)
(b) \( ^{n+2}C_r \)
(c) \( ^{n+2}C_{r+1} \)
(d) \( ^{n+2}C_{r+2} \)
Answer: (c) \( ^{n+2}C_{r+1} \)

COMMITTE

Question. From a council of 7 Hindus, 5 Muslims and 4 Christians a committee of 5 is to be formed with 1 christian and at least two Hindus. The number of ways of forming the committee is
(a) 1220
(b) 1240
(c) 1680
(d) 1280
Answer: (c) 1680

Question. An examination paper, which is divided into two groups consisting of 3 and 4 questions respectively carries the note it is not necessary to answer all the questions. One question atleast should be answered form each group. The number of ways can an examinee select the questions is
(a) 22
(b) 105
(c) \( ^3P_3 \times ^4P_4 \)
(d) \( ^3C_3 \times ^4C_4 \)
Answer: (b) 105

Question. A guard of 12 men is formed form n soldiers. If A and B are three times as often together on guard as C, D, E then =
(a) 16
(b) 32
(c) 64
(d) 8
Answer: (b) 32

Question. A quesion paper consisting of 10 questions is divided into 3 parts with 5, 3, 2 quesions. A candidate is to answer 6 quesions without neglecting any part. The number of ways in which he can make up his choice is
(a) 175
(b) 200
(c) 225
(d) 150
Answer: (a) 175

GEOMETRICAL APPLICATIONS

Question. The maximum number of points of intersection of 8 straight lines is
(a) 8
(b) 16
(c) 28
(d) 56
Answer: (c) 28

Question. The maximum number of points of intersection of 8 circles is
(a) 16
(b) 24
(c) 28
(d) 56
Answer: (d) 56

Question. There are 10 points in a plane and A is one of them. If no three of the points are collinear then the number of triangles formed with A as vertex is
(a) 45
(b) 36
(c) 84
(d) 120
Answer: (b) 36

Question. Six points are taken on a circle. The number of triangles formed inside the circle is
(a) 20
(b) 22
(c) 25
(d) 32
Answer: (a) 20

DISTRIBUTION OF DISSIMILAR THINGS

Question. The number of ways in which a pack of 52 cards of four different suits be distributed equally among 4 players so that each may have ace, king, queen and knave of the same suit is
(a) \( \frac{4!(36!)}{(9!)^4} \)
(b) \( \frac{36!}{(9!)^4} \)
(c) \( \frac{2(36!)}{(9!)^4} \)
(d) \( \frac{3(36!)}{(9!)^4} \)
Answer: (a) \( \frac{4!(36!)}{(9!)^4} \)

Question. A shop keeper sells three varieties of perfumes and he has a large number of bottles of the same size of each variety in this stock. there are 5 places in a row in his showcase. The number of different ways of displaying the three varieties of perfumes in the showcase is
(a) 6
(b) 50
(c) 150
(d) 60
Answer: (c) 150

DISTRIBUTION OF SIMILAR THINGS

Question. The number of ways of distiributing 8 identical balls in three distinct boxes so that none of the boxes is empty
(a) 5
(b) \( ^8C_3 \)
(c) \( 3^8 \)
(d) 21
Answer: (d) 21

Question. The number of ways of selecting 8 books from a library which has 9 books each of Mathematics, Physics, Chemistry and English is
(a) 165
(b) \( ^{27}C_8 \)
(c) \( 3^8 \)
(d) 81
Answer: (a) 165

Question. The number of ways in which an examiner can assign 30 marks to 8 questions giving not less than 2 marks to any questions is
(a) \( ^{19}C_7 \)
(b) \( ^{20}C_7 \)
(c) \( ^{21}C_7 \)
(d) \( ^{22}C_7 \)
Answer: (c) \( ^{21}C_7 \)

Question. If a, b, c are three natural numbers in AP and a + b + c = 21 then the possible number of ordered triplets (a, b, c) is
(a) 15
(b) 14
(c) 13
(d) 12
Answer: (c) 13

Question. There are 15 trees in a row. 4 trees are to be cut down. The number of ways that no two of the cut down trees are consecutive is
(a) \( ^{11}C_4 \)
(b) \( ^{12}C_4 \)
(c) \( ^{15}C_4 \)
(d) \( ^{14}C_4 \)
Answer: (b) \( ^{12}C_4 \)

MISCELLANEOUS PROBLEMS

Question. If \( 7^r \) divides \( 55! \) then \( r = \)
(a) 6
(b) 7
(c) 8
(d) 9
Answer: (c) 8

Question. Largest value of \( n \), so that \( 10^n \) divides \( 51! \)
(a) 47
(b) 12
(c) 35
(d) 59
Answer: (b) 12

Question. The number of ways of writing 98 as the product of two positive integers is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (c) 3

Question. Out of 5 apples, 10 mangoes and 15 oranges, the number of ways of distributing 15 fruits each to two persons is
(a) 56
(b) 64
(c) 66
(d) 72
Answer: (c) 66

Question. The number of ways in which four letters can be selected from the letters of the word “MATHEMATICS” is
(a) 133
(b) 146
(c) 136
(d) 73
Answer: (c) 136

Question. \( 3^r \) divides \( 50! \) then maximum value of r =
(a) 27
(b) 25
(c) 23
(d) 22
Answer: (d) 22

Question. The number of ways in which 5 identical balls can be kept in 10 identical boxes, if not more than one can go into a box, is
(a) \( ^{10}P_5 \)
(b) \( \binom{10}{5} \)
(c) 5
(d) 1
Answer: (d) 1

Question. The number of rational numbers lying in the interval (2002, 2003) all of whose digits after the decimal point are non-zero and are in decreasing order is
(a) \( 2^6 \)
(b) \( 2^9 \)
(c) \( 2^9 - 1 \)
(d) \( 2^{10} - 1 \)
Answer: (c) \( 2^9 - 1 \)

Question. If the set S= {1, 2, 3, ...., 12} is to be partitioned into three sets A, B, C of equal size such that \( A \cup B \cup C = S \), \( A \cap B = B \cap C = A \cap C = \phi \) then the number of ways of partitioning S is
(a) \( \frac{12!}{(3!)^4} \)
(b) \( \frac{12!}{3!(4!)^3} \)
(c) \( \frac{12!}{3!(3!)^4} \)
(d) \( \frac{12!}{(4!)^3} \)
Answer: (d) \( \frac{12!}{(4!)^3} \)

Question. Two teams are to play a series of 5 matches between them. A match ends in a win or loss or draw for a team. A number of people forecast the result of each match and no two people make the same forecast for the series of matches. The smallest group of people in which one person forecast correctly for all the matches will contain ‘n’ people, where n =
(a) 81
(b) 243
(c) 486
(d) 144
Answer: (b) 243

Question. 18 guests have to be seated half on each side of a long table. 4 particular guests desire to sit on one particular side and 3 others on the other side. Then the number of ways in which the sitting arrangements can be made
(a) \( (9!)^2 \)
(b) \( ^{11}C_4(9!)^2 \)
(c) \( ^{11}C_3(9!)^2 \)
(d) \( ^{11}C_5(9!)^2 \)
Answer: (d) \( ^{11}C_5(9!)^2 \)

Question. If there are 5 periods in each working day of a school, then the number of ways that you can arrange 4 subjects during the working day is
(a) 220
(b) 240
(c) 260
(d) 280
Answer: (b) 240

Question. The number of positive integral solutions of \( x^2 - y^2 = 352706 \) is
(a) 2
(b) 1
(c) 0
(d) 3
Answer: (c) 0

Question. The number of different combinations that can be formed out of the letters of the word 'INFINITE' taken four at a time is
(a) 20
(b) 22
(c) 24
(d) 120
Answer: (b) 22

Question. Eight chairs are numbered 1 to 8. Two women and three men wish to occupy one chair each. First the women choose the chairs from among the chairs marked 1 to 4, then the men select the chairs from among the remaining. The number of possible arrangements is
(a) \( ^4P_2 \cdot ^6P_3 \)
(b) \( ^6C_3 \cdot ^4C_2 \)
(c) \( ^4C_2 \cdot ^4P_3 \)
(d) \( ^4P_2 \cdot ^4P_3 \)
Answer: (a) \( ^4P_2 \cdot ^6P_3 \)

Question. There are 5 English, 4 Sanskrit and 3 Telugu books. Two books from each group are to be arranged in a shelf. The number of possible arrangements is
(a) (180) 6!
(b) (12) 7!
(c) 7!
(d) 180
Answer: (a) (180) 6!

Question. The number of ways in which a TRUE or FALSE examination of n statements can be answered on the asumption that no two consecutive questions are answered the same way is
(a) \( 2^{n-1} \)
(b) \( 2^n \)
(c) 1
(d) 2
Answer: (d) 2

Question. Three ladies have each brought their one child for admission to a school. The principal wants to interview the six persons one by one subject to the condition that no mother is interviewed before her child. The number of ways in which interviews can be arranged is
(a) 6
(b) 36
(c) 72
(d) 90
Answer: (d) 90

Question. A delegation of four friends are to be selected from a group of 12 friends. The number of ways the delegation be selected if two particular friends refused to be together and two other particular friends wish to be together only in the delegation.
(a) 226
(b) 114
(c) 156
(d) 170
Answer: (a) 226

Question. A class contains 4 boys and ‘g’ girls. Every sunday five students, including at least three boys go for a picnic to Zoo Park, a different group being sent every week. During, the picnic, the class teacher gives each girl in the group a doll. If the total number of dolls distributed was 85, then value of ‘g’ is
(a) 15
(b) 12
(c) 8
(d) 5
Answer: (d) 5

Question. The number of rectangles excluding squares from a rectangle of size 9 x 6 is
(a) 391
(b) 791
(c) 842
(d) 250
Answer: (b) 791

Question. In the next world cup of cricket there will be 12 teams, divided equally in two groups. Teams of each group will play a match against each other. From each group 3 top teams will qualify for the next round. In this round each team will play against other once. Each of the four top teams of this round will play a match against the other three. Two top teams of this round will go to the final round, where they will play the best of three matches. The minimum number of matches in the next world cup will be
(a) 54
(b) 53
(c) 38
(d) 55
Answer: (b) 53