Class 11 Mathematics Combinations MCQs Set 02

Question. In a plane there are 37 straight lines of which 13 pass through the point A and 11 pass through the point B. Besides, no three lines pass through one point, no line passes through both point A and B, and no two are parallel. Then the number of points of intersection of the lines is
(a) 535
(b) 601
(c) 728
(d) 600
Answer: (a) 535

Question. A regular polygon of \(n\) sides has 170 diagonals. Then \(n =\)
(a) 12
(b) 17
(c) 20
(d) 25
Answer: (c) 20

Question. If \(m\) parallel lines in a plane are intersected by \(n\) parallel lines then number of parallelograms formed is
(a) \(\frac{m!n!}{(2!)^2}\)
(b) \(\frac{m!n!}{(m-2)!(n-2)!}\)
(c) \(\frac{m!n!}{(2!)^2(m-2)!(n-2)!}\)
(d) \(\frac{(m+n)!}{(m+n-2)!2!}\)
Answer: (c) \(\frac{m!n!}{(2!)^2(m-2)!(n-2)!}\)

Question. The number of rectangles which are not squares in a chess board is
(a) \(^8C_2 \times ^8C_2 - 8^2\)
(b) \((^8C_2 \times ^8C_2) - \sum 8^2\)
(c) \(^9C_2 \times ^9C_2 - 8^2\)
(d) \((^9C_2 \times ^9C_2) - \sum 8^2\)
Answer: (d) \((^9C_2 \times ^9C_2) - \sum 8^2\)

Question. The number of ways of selecting two squares \((1 \times 1)\) in a chess board such that they have a side in common is
(a) 224
(b) 112
(c) 56
(d) 68
Answer: (b) 112

TOTAL NUMBER OF COMBINATIONS

Question. A basket contains 4 Oranges, 5 Apples, 6 Mangoes. The number of ways a person make selection of fruits from the basket is
(a) 209
(b) 210
(c) 211
(d) 212
Answer: (a) 209

Question. No. of ways of selecting none or more of 10 identical things is
(a) 1
(b) 10
(c) 11
(d) 9
Answer: (c) 11

Question. The no of ways of selecting atleast one letter from the letters of the word “PROPORTION” is.
(a) 288
(b) 286
(c) 289
(d) 287
Answer: (d) 287

Question. The number of different products that can be formed with 8 prime numbers is
(a) 247
(b) 252
(c) 5
(d) 248
Answer: (a) 247

Question. At an election, a voter may vote for any number of candidates, not greater than the number to be elected. There are 10 candidates and 4 are to be elected. If a voter votes for atleast one candidate, then the number of ways in which he can vote, is
(a) 385
(b) 1110
(c) 5040
(d) 6210
Answer: (a) 385

DISTRIBUTION OF DISSIMILAR THINGS INTO GROUPS

Question. No. of ways of dividing 80 cards into 5 equal groups of 16 each is
(a) \(\frac{80!}{(16!)^5}\)
(b) \(\frac{80!}{(5!)^{16}}\)
(c) \(\frac{80!}{(5!)^5}\)
(d) \(\frac{80!}{(16!)^5 \cdot 5!}\)
Answer: (d) \(\frac{80!}{(16!)^5 \cdot 5!}\)

Question. The number of ways a pack of 52 cards can be divided among four players in 4 sets, three of them having 17 cards each and the fourth one just 1 card is
(a) \(\frac{52!}{(17!)^3} \times 4\)
(b) \(\frac{52!}{3 \cdot (17!)^3}\)
(c) \(\frac{52!}{3! (17!)^3}\)
(d) \(\frac{52!}{3!^3 (17!)}\)
Answer: (a) \(\frac{52!}{(17!)^3} \times 4\)

Question. The number of ways can 5 things be divided between A and B so that each receive at least one thing is
(a) 30
(b) 60
(c) 20
(d) 80
Answer: (a) 30

DISTRIBUTION OF SIMILAR THINGS INTO GROUPS

Question. In how many ways can 3 sovereigns be given away when there are 4 applicants and any applicant may have either 0, 1, 2 or 3 sovereigns?
(a) 15
(b) 20
(c) 24
(d) 48
Answer: (b) 20

Question. The number of ways of distributing 15 things to 4 persons each receiving atleast two is
(a) 120
(b) 60
(c) 28
(d) 108
Answer: (a) 120

Question. The number of non-negative integral solutions of \(x_1 + x_2 + x_3 + x_4 \leq n\) (where \(n\) is a non negative integer) is
(a) \(^{n+3}C_3\)
(b) \(^{n+5}C_5\)
(c) \(^{n+4}C_4\)
(d) \(^{n+6}C_6\)
Answer: (c) \(^{n+4}C_4\)

NUMBER OF DIVISORS

Question. The number of odd divisors of 128 is
(a) 8
(b) 7
(c) 0
(d) 1
Answer: (d) 1

Question. The number of odd proper positive divisors of \(3^a 6^b 21^c\) is \((a, b, c \in \mathbb{N})\)
(a) \((a + 1) (b + 1) (c + 1) - 2\)
(b) \((a + b + c + 1)(c + 1) - 1\)
(c) \((a + 1)(b + 1)(c + 1) - 1\)
(d) \((a + 1)(b + 1)(c + 1)\)
Answer: (b) \((a + b + c + 1)(c + 1) - 1\)

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

APPLICATIONS OF ONTO FUNCTION

Question. 5 balls of different colours are to be kept in 3 boxes of different sizes. Each box can hold all five balls. Number of ways in which the balls can be kept in the boxes so that no box remain empty is
(a) 60
(b) 90
(c) 150
(d) 200
Answer: (c) 150

Question. If \( ^nC_3 = {}^nC_9 \) then \( ^nC_2 = \)
(a) 66
(b) 132
(c) 72
(d) 98
Answer: (a) 66

Question. If \( ^nC_3 : {}^{2n-1}C_2 = 8:15 \) then 'n' is
(a) 5
(b) 6
(c) 8
(d) 7
Answer: (c) 8

Question. If \( ^nC_{r-1} = 36 \), \( ^nC_r = 84 \), \( ^nC_{r+1} = 126 \) then (n, r) =
(a) (9, 6)
(b) (9, 5)
(c) (9, 3)
(d) (9, 2)
Answer: (c) (9, 3)

Question. The least value of the natural number 'n' satisfying \( c(n,5) + c(n,6) > c(n+1,5) \)
(a) 10
(b) 12
(c) 13
(d) 11
Answer: (d) 11

Question. \( ^{22}C_5 + \sum_{i=1}^{4} {}^{(26-i)}C_4 = \)
(a) \( ^{27}C_5 \)
(b) \( ^{27}C_4 \)
(c) \( ^{26}C_4 \)
(d) \( ^{26}C_5 \)
Answer: (d) \( ^{26}C_5 \)

Question. If n and r are integers such that 1 < r < n then n . C (n-1, r-1) =
(a) C (n, r)
(b) n . C (n, r)
(c) r C (n, r)
(d) (n - 1) . C (n, r)
Answer: (c) r C (n, r)

SELECTION OF DISSIMILAR THINGS

Question. A committee of 5 is to be formed from 6 boys and 5 girls. The number of ways that the committee can be formed so that the committee contains atleast one boy and one girl having majority of boys is
(a) 240
(b) 245
(c) 250
(d) 275
Answer: (d) 275

Question. The number of different ways in which a committee of 4 members formed out of 6 Asians, 3 Europeans and 4 Americans if the committee is to have atleast one from each of the 3 regional groups is
(a) 320
(b) 340
(c) 360
(d) 380
Answer: (c) 360

Question. 153 games were played at a chess tournament with each contestant playing once against each of the others. The number of participants is
(a) 16
(b) 17
(c) 18
(d) 19
Answer: (c) 18

Question. There were two women participating in a chess tournament. Every participant played two games with the other participants. the number of games that the men played between themselves proved to exceed by 66 the number of games that the men played with the women. The number of participants is
(a) 6
(b) 11
(c) 13
(d) 10
Answer: (c) 13

Question. In how many ways can 12 boys be seated on two benches x, y; 6 on each bench if two of them A,B are to sit on bench x and C,D on the bench y
(a) \( ^8C_4 \cdot 6! \cdot 6! \)
(b) \( ^8C_6 \cdot 6! \cdot 6! \)
(c) \( 6! \cdot 6! \)
(d) \( ^8C_4 \cdot 6! \)
Answer: (a) \( ^8C_4 \cdot 6! \cdot 6! \)

Question. If 'n' different things are arranged around a circle then the number of ways of selecting 3 objects when no two selected objects are together
(a) \( \frac{(n-4)(n-5)}{3!} \)
(b) \( \frac{n(n-4)(n-5)}{3!} \)
(c) \( \frac{n(n-5)}{6} \)
(d) \( \frac{n(n-4)}{3} \)
Answer: (b) \( \frac{n(n-4)(n-5)}{3!} \)

PROBLEMS ON INCLUDING & EXCLUDING

Question. Out of 7 men and 4 women a committee of 5 is to be formed. The number of ways in which this can be done so as to include atleast 2 women is
(a) 210
(b) 301
(c) 294
(d) 84
Answer: (b) 301

Question. A box contains 2 white, 3 black and 4 red balls. (Balls are of different sizes). In how many ways can 3 balls be drawn from the box if atleast one black ball is to be included in the draw is
(a) 84
(b) 64
(c) 60
(d) 120
Answer: (b) 64

Question. The number of ways that a volleyball team of 6 players can be selected out of 10 players so that 2 particular players are excluded is
(a) 56
(b) 55
(c) 27
(d) 28
Answer: (d) 28

Question. There are 10 balls of different colours. In how many ways is it possible to select 7 of them so as to exclude the white and the black ball ?
(a) 8
(b) 7
(c) 16
(d) 20
Answer: (a) 8

Question. If 20% of three element subsets (i.e. subsets containing exactly three elements) of the set \( A = \{a_1, a_2, \ldots, a_n\} \) contain \( a_1 \), then value of 'n' is
(a) 15
(b) 16
(c) 17
(d) 18
Answer: (a) 15

GEOMETRICAL APPLICATIONS

Question. There are 15 lines terminating at a point. The number of angles formed is
(a) 105
(b) 120
(c) 125
(d) 120
Answer: (a) 105

Question. Given five line segments of length 2, 3, 4, 5, 6 units. Then the number of triangles that can be formed by joining these lines is
(a) \( ^5C_3 - 3 \)
(b) \( ^5C_3 - 1 \)
(c) \( ^5C_3 \)
(d) \( ^5C_3 - 2 \)
Answer: (a) \( ^5C_3 - 3 \)

Question. The number of triangles whose vertices are at the vertices of an octagon but none of whose sides happen to come from the sides of the octagon is
(a) 24
(b) 52
(c) 48
(d) 16
Answer: (d) 16