Class 11 Mathematics Combinations MCQs Set 01

Question. If \(r>1\) then \(\frac{^nP_r}{^nC_r}\) is
(a) is an integer
(b) may be fraction
(c) is an odd number
(d) an even number
Answer: (d) an even number

Question. The value of \(1 \times 3 \times 5 \dots (2n-1) 2^n =\)
(a) \(\frac{(2n)!}{2^n}\)
(b) \(\frac{(2n)!}{n!}\)
(c) \(\frac{n!}{(2n)!}\)
(d) \(2n\)
Answer: (b) \(\frac{(2n)!}{n!}\)

Question. If \(^nC_r + ^nC_{r+1} = ^{n+1}C_x\) then \(x =\)
(a) \(r\)
(b) \(r - 1\)
(c) \(n\)
(d) \(r + 1\)
Answer: (d) \(r + 1\)

Question. The least value of \(n\) so that \(^nC_6 + ^nC_7 > ^{n+1}C_6\) is
(a) 13
(b) 12
(c) 11
(d) 10
Answer: (a) 13

Question. No. of ways of selecting 2 girls and 3 boys from 3 girls and 5 boys is
(a) 20
(b) 24
(c) 30
(d) 48
Answer: (c) 30

Question. A father with 6 children takes 3 at a time to a park without taking the same children. How often father goes to the park?
(a) 14
(b) 16
(c) 18
(d) 20
Answer: (d) 20

Question. From 15 players the number of ways of selecting 6 so as to exclude a particular player is
(a) \(^{14}C_5\)
(b) \(^{15}C_6\)
(c) \(^{15}C_5\)
(d) \(^{14}C_6\)
Answer: (d) \(^{14}C_6\)

Question. The number of three digit numbers of the form \(xyz\) where \(x>y>z\) is
(a) 120
(b) 720
(c) 600
(d) 100
Answer: (c) 600

Question. A set contains \((2n + 1)\) elements. If the number of subsets of this set which contain atmost ‘\(n\)’ elements is 4096, then the value of \(n\) is
(a) 6
(b) 15
(c) 21
(d) 20
Answer: (a) 6

Question. In a library there are \((2n+1)\) books. If a student selects atleast \((n+1)\) books in 256 ways then the number of books in the library is
(a) 7
(b) 8
(c) 9
(d) 6
Answer: (c) 9

Question. A set contains \((2n+1)\) elements. The number of subsets of this set which contains more than '\(n\)' elements is
(a) \(2^{n-1}\)
(b) \(2^n\)
(c) \(2^{n+1}\)
(d) \(2^{2n}\)
Answer: (d) \(2^{2n}\)

Question. The number of subsets of the set \(A = \{1, 2, 3, \dots, 9\}\) containing at least one odd number is
(a) 324
(b) 396
(c) 496
(d) 512
Answer: (c) 496

Question. The number of ways in which 52 cards can be divided into 4 sets of 13 each is
(a) \(\frac{52!}{(13!)^4}\)
(b) \(\frac{52!}{4!(13!)^4}\)
(c) \(\frac{52!}{4^{13}}\)
(d) \(\frac{52!}{13! \, 4^{13}}\)
Answer: (b) \(\frac{52!}{4!(13!)^4}\)

Question. No. of ways of distributing \((p + q + r)\) different things to 3 persons so that one person gets \(p\) things, 2nd person \(q\) things 3rd person \(r\) things is
(a) \(\frac{(p+q+r)!}{p! \cdot q! \cdot r!} \times 3!\)
(b) \(\frac{(p+q+r)!}{p! \cdot q! \cdot r!}\)
(c) \(\frac{(p+q+r)!}{3! \cdot p! \cdot q! \cdot r!}\)
(d) \(p! \cdot q! \cdot r!\)
Answer: (a) \(\frac{(p+q+r)!}{p! \cdot q! \cdot r!} \times 3!\)

Question. The number of quadratic expressions with the coefficients drawn from the set \(\{1, 2, 3, 4\}\) is
(a) 27
(b) 36
(c) 64
(d) 48
Answer: (c) 64

Question. The number of ways in which a man can invite one or more of his 8 friends to dinner is
(a) 28
(b) 128
(c) 240
(d) 255
Answer: (d) 255

Question. The no.of 5digit 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

Question. No. of ways of selecting 3 consecutive objects out of 15 objects (distinct) placed around a circle is
(a) \(^{15}C_r\)
(b) \(^{14}C_3\)
(c) 13
(d) 15
Answer: (d) 15

Question. If \(^{15}C_{3r} = ^{15}C_{r+3}\) then \(r =\)
(a) \(\frac{3}{2}\)
(b) 3
(c) 4
(d) 5
Answer: (b) 3

Question. If \(^{2n}C_3 : ^nC_2 = 44 : 3\) then \(n =\)
(a) 6
(b) 7
(c) 8
(d) 9
Answer: (a) 6

Question. If \(^{n+2}C_8 : ^{n-2}P_4 = 57 : 16\) then \(n =\)
(a) 17
(b) 18
(c) 19
(d) 20
Answer: (c) 19

Question. \(^{14}C_4 + \sum_{j=1}^{4} (18-j)C_3 =\)
(a) \(^{14}C_5\)
(b) \(^{18}C_5\)
(c) \(^{18}C_4\)
(d) \(^{19}C_4\)
Answer: (c) \(^{18}C_4\)

Question. If \(^nP_r = 30240\) and \(^nC_r = 252\) then the ordered pair \((n,r) =\)
(a) \((12,6)\)
(b) \((10,5)\)
(c) \((9,4)\)
(d) \((16,7)\)
Answer: (b) \((10,5)\)

Question. If C (2n, 3) : C (n, 2) = 12 : 1, then \(n =\)
(a) 4
(b) 5
(c) 6
(d) 8
Answer: (b) 5

SELECTION OF DISSIMILAR THINGS

Question. A group contains 6 men and 3 women. A committee is to be formed with 5 people containing 3 men and 2 women. The number of different committees that can be formed is
(a) \(^9C_5\)
(b) \(^6C_3 \times ^3C_2\)
(c) \(^6C_3\)
(d) \(^3C_2\)
Answer: (b) \(^6C_3 \times ^3C_2\)

Question. A team of 11 players has to be choosen from the groups consisting of 6 and 8 players respectively. The number of ways of selecting them so that each selection contains atleast 4 players from the first group is
(a) 120
(b) 280
(c) 344
(d) 244
Answer: (c) 344

Question. A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is
(a) 196
(b) 280
(c) 346
(d) 140
Answer: (a) 196

Question. Ten students are participating in a race. The number of ways in which the first three places can be taken as
(a) 3
(b) \(^{10}C_3\)
(c) \(^{10}P_3\)
(d) \(^{10}P_4\)
Answer: (c) \(^{10}P_3\)

Question. In a chess tournment, where the participants were to play one game with another. Two chess players fell ill, having played 3 games each. If the total number of games played is 84, the number of participants at the beginning was
(a) 15
(b) 16
(c) 20
(d) 21
Answer: (a) 15

Question. From a company of 20 soldiers any 5 are placed on guard, each batch to watch 5 hours. For what length of time in hours can different batches be selected is
(a) \(^{29}C_5\)
(b) \(^{20}P_5\)
(c) \(^{20}C_5 \times 5\)
(d) \(^{20}P_5 \times 5\)
Answer: (c) \(^{20}C_5 \times 5\)

Question. 10 persons are seated at round table. The number of ways of selecting 3 persons out of them if no two persons are adjacent to each other is
(a) 50
(b) 62
(c) 56
(d) 57
Answer: (a) 50

PROBLEMS ON INCLUDING & EXCLUDING

Question. The number of ways in which a team of 11 players can be selected from 22 players including 2 of them and excluding 4 of them is
(a) \(^{16}C_6\)
(b) \(^{16}C_7\)
(c) \(^{16}C_8\)
(d) \(^{20}C_7\)
Answer: (b) \(^{16}C_7\)

Question. The number of ways that a volley ball 6 can be selected out of 10 players so that 2 particular players are included is
(a) 72
(b) 70
(c) 68
(d) 66
Answer: (b) 70

Question. In a shelf there are 10 English and 8 Telugu books. The number of ways in which 6 books can be chosen if a particular English book is excluded and a particular Telugu book is excluded is
(a) \(^9C_3 \cdot ^7C_3\)
(b) \(^{16}C_6\)
(c) \(^9C_3 \cdot ^8C_3\)
(d) \(^{18}C_8\)
Answer: (b) \(^{16}C_6\)

Question. The number of permutations of \(n\) things taken \(r\) at a time if 3 particular things always occur is
(a) \(\frac{(n-3)!}{(n-r)!} r(r-1)(r-2)\)
(b) \(\frac{(n-3)!}{(r-3)!}\)
(c) \(\frac{(n-3)!}{(n-r)!} \times 3\)
(d) \(\frac{(n-3)!}{(r-2)!}\)
Answer: (a) \(\frac{(n-3)!}{(n-r)!} r(r-1)(r-2)\)

Question. The number of all three element subsets of the set \(\{a_1, a_2, a_3, \dots, a_n\}\) which contain \(a_3\) is
(a) \(^nC_3\)
(b) \(^{n-1}C_3\)
(c) \(^{n-1}C_2\)
(d) \(^nC_2\)
Answer: (c) \(^{n-1}C_2\)

GEOMETRICAL APPLICATIONS

Question. There are 10 straight lines in a plane no two of which are parallel and no three are concurrent. The points of intersection are joined, then the no. of fresh lines formed are
(a) 630
(b) 615
(c) 730
(d) 600
Answer: (a) 630

Question. If a line segment be cut at ‘\(n\)’ points, then the number of line segments formed is
(a) \(n(n+3)\)
(b) \(\frac{n(n-3)}{2}\)
(c) \(\frac{(n+2)(n+1)}{2}\)
(d) \(n\)
Answer: (c) \(\frac{(n+2)(n+1)}{2}\)

Question. The sides AB, BC, CA of a triangle ABC have 3, 4 and 5 interior points respectively on them. The number of triangles that can be formed using these points as vertices is
(a) 205
(b) 220
(c) 225
(d) 230
Answer: (a) 205

Question. There are 'p' points in space of which 'q' points are coplanar. Then the number of planes formed is
(a) \(^pC_3 - ^qC_3\)
(b) \(^pC_3 - ^qC_3 + 1\)
(c) \(^pC_2 - ^qC_2\)
(d) \(^pC_3 - ^qC_2\)
Answer: (b) \(^pC_3 - ^qC_3 + 1\)