Class 11 Mathematics Combinations MCQs Set 05

Question. A boy has 3 library tickets and 8 books of his interest in the library. Out of these 8, he does not want to borrow Chemistry part II, unless Chemistry part I is also borrowed. The number of ways in which he can choose the three books to be borrowed is
(a) 41
(b) 32
(c) 51
(d) 26
Answer: (a) 41

Question. Let x.y.z=105 where x, y, z \( \in \) N. Then number of ordered triplets (x, y, z) satisfying the given equation is
(a) 15
(b) 27
(c) 6
(d) 33
Answer: (b) 27

Question. The no. of ways can a group of 5 letters be formed out of 5a 's, 5b 's, 5c 's and 5d 's is
(a) \( ^5C_4 \times 5 \)
(b) \( ^5C_4 \times 5 \)
(c) \( ^9C_5 \)
(d) \( ^8C_5 \)
Answer: (d) \( ^8C_5 \)

Question. A is a set containing n elements. A subset P of A is chosen.The set A is reconstructed by replacing the elements of P. A subset Q of A is again chosen. The number of ways of choosing P and Q so that P \( \cap \) Q contains exactly two elements is
(a) \( 9 \cdot ^nC_2 \)
(b) \( 3^n - ^nC_2 \)
(c) \( 2 \cdot ^nC_n \)
(d) \( ^nC_2 \cdot 3^{n-2} \)
Answer: (d) \( ^nC_2 \cdot 3^{n-2} \)

Question. There are 10 bags \( B_1, B_2, B_3, \dots, B_{10} \) which contain 21, 22, ...., 30 different articles respectively. The total number of ways to bring out 10 articles from any one bag is
(a) \( ^{31}C_{20} - ^{21}C_{10} \)
(b) \( ^{31}C_{21} \)
(c) \( ^{31}C_{20} \)
(d) \( ^{32}C_{20} \)
Answer: (a) \( ^{31}C_{20} - ^{21}C_{10} \)

Question. A Tennis tournament is to be played by 10 pairs of students and each pair is to play with every other pair one set. If four sets are played each day then the number of days should be allowed for the tournament is
(a) 12
(b) 16
(c) 80
(d) 90
Answer: (a) 12

Question. The results of 21 football matches (win, lose, draw) are to be predicted. The no. of different forecasts that can contain 19 wins is
(a) 210
(b) 640
(c) 840
(d) 1260
Answer: (c) 840

Question. n bit strings are made by filling the digits 0 or 1. The number of strings in which there are exactly k zeros with no two 0's consecutive is
(a) \( ^{n-k}C_k \)
(b) \( ^{n-k+1}C_k \)
(c) \( ^{n-k-1}C_k \)
(d) \( ^{n-k}C_{k-1} \)
Answer: (b) \( ^{n-k+1}C_k \)

Question. The interior angles of a regular polygon measure \( 120^0 \) each. The number of diagonal of the polygon is
(a) 8
(b) 9
(c) 18
(d) 10
Answer: (b) 9

Question. The number of ways in which we can select 4 numbers from 1 to 30 so as to exclude every selection of four consecutive numbers is
(a) 27378
(b) 27405
(c) 27504
(d) 27387
Answer: (a) 27378

Question. Given that ‘n’ is odd, the number of ways in which three numbers in A.P can be selected from 1, 2, 3, .... ,n is
(a) \( (n - 1)^2 / 2 \)
(b) \( (n + 1)^2 / 4 \)
(c) \( (n + 1)^2 / 2 \)
(d) \( (n - 1)^2 / 4 \)
Answer: (d) \( (n - 1)^2 / 4 \)

Question. A child attempts to open a five disc-lock (each disc consists of digits {0, 1, ...., 9}) He takes 5 sec time to dial a particular number on the disc. If he does so for 5 hrs. every day, then the number of days he would be sure to open the lock is
(a) 30
(b) 28
(c) 27
(d) 25
Answer: (b) 28

Question. In a plane there are two families of lines \( y = x + r, y = -x + r \), where \( r \in \{0, 1, 2, 3, 4\} \). The number of squares of diagonals of the length 2 units formed by the lines is
(a) 9
(b) 16
(c) 25
(d) 36
Answer: (a) 9

Question. The number of ordered pairs \( (m, n), m, n \in \{1, 2, \dots, 100\} \) such that \( 7^m + 7^n \) is divisible by 5 is
(a) 1250
(b) 2000
(c) 2500
(d) 5000
Answer: (c) 2500

Question. The number of different words which can be formed by taking 4 letters at a time out of the letters of the word 'EXPRESSION' is
(a) 2090
(b) 2190
(c) 2454
(d) 2354
Answer: (b) 2190

Question. The number of permutations of the letters of the word 'INDEPENDENT' taken 5 at a time is
(a) 3302
(b) 3320
(c) 3230
(d) 3203
Answer: (b) 3320

Question. The number of ways that the letters of the word "PERSON" can be placed in the squares of the adjoining figure so that no row remains empty
R1 - [ ] [ ]
R2 - [ ] [ ]
R3 - [ ] [ ] [ ] [ ]
(a) 20x6!
(b) 26x6!
(c) 20x5!
(d) 26x5!
Answer: (b) 26x6!

Question. A rectangle with sides 2m -1, 2n - 1 is divided into squares of unit length by drawing parallel lines as shown in the diagram.
The number of rectangles with odd side length is
(a) \( (m + n + 1)^2 \)
(b) \( mn(m + 1)(n + 1) \)
(c) \( m^2 n^2 \)
(d) \( 4^{m+n-1} \)
Answer: (c) \( m^2 n^2 \)

Question. The number of positive integral divisors of 1200 which are multiples of ‘6’ is
(a) 6
(b) 12
(c) 8
(d) 24
Answer: (b) 12

Question. The sum of positive integral divisors of 600 which are multiples of 10 is
(a) 840
(b) 3360
(c) 1680
(d) 420
Answer: (c) 1680

Question. The number of positive integral solution sets of the equation \( xyz = 150 \) is
(a) 27
(b) 54
(c) 108
(d) 9
Answer: (b) 54

Question. In how many ways 3 boys and 15 girls can sit together in a row such that between any 2 boys there is atleast 2 girls
(a) \( ^{14}C_3 \cdot 3! \cdot 15! \)
(b) \( ^{14}C_3 \cdot 3! \cdot 10! \)
(c) \( ^{18}C_3 \cdot 3! \cdot 15! \)
(d) \( ^{12}C_3 \cdot 3! \cdot 15! \)
Answer: (a) \( ^{14}C_3 \cdot 3! \cdot 15! \)

Question. A test has 4 parts. The first 3 parts carry 10 marks each and the 4th part carries 20 marks. Assuming that marks are not given in fractions, then the number of ways in which a candidate can get 30 marks out of 50 is
(a) 1011
(b) 1111
(c) 999
(d) 1001
Answer: (b) 1111

Question. In an examination of 9 papers a candidate has to pass in more papers than the number of papers in which he fails in order to be successful. The number of ways in which he can be unsuccessful is
(a) 255
(b) 256
(c) 193
(d) 319
Answer: (b) 256

Question. The sum \( S = \frac{1}{9!} + \frac{1}{3!7!} + \frac{1}{5!5!} + \frac{1}{7!3!} + \frac{1}{9!} = \)
(a) \( 2^9 / 10! \)
(b) \( 2^{10} / 8 \)
(c) \( 2^{11} / 9! \)
(d) \( \frac{2^{12}}{12!} \)
Answer: (a) \( 2^9 / 10! \)

Question. The number of different triangles formed by joining the points A, B, C, D, E, F and G as shown in the figure given below is
A -- B -- C -- D -- E
          |
          F
          |
          G
(a) 20
(b) 24
(c) 25
(d) 30
Answer: (b) 24

Question. Let A and B be two sets containing 2 elements and 4 elements respectively. The number of subsets of \( A \times B \) having 3 or more elements is
(a) 220
(b) 219
(c) 211
(d) 256
Answer: (b) 219

Question. How many different words can be formed by jumbling the letters in the word ‘MISSISSIPPI’ in which no two S are adjacent?
(a) \( 7 \cdot ^6C_4 \cdot ^8C_4 \)
(b) \( 8 \cdot ^6C_4 \cdot ^7C_4 \)
(c) \( 6 \cdot 7 \cdot ^8C_4 \)
(d) \( 6 \cdot 8 \cdot ^7C_4 \)
Answer: (a) \( 7 \cdot ^6C_4 \cdot ^8C_4 \)

Question. The number of ways of selecting 3-member subset of \( \{1, 2, 3, \dots, 25\} \) so that the numbers form a G.P. with integer common ratio is
(a) 10
(b) 11
(c) 12
(d) 15
Answer: (a) 10

Question. The number of n digit numbers, which contain the digits 2 and 7, but not the digits 0, 1, 8, 9.
(a) \( 6^n - 5^n - 5^n + 4^n \)
(b) \( 6^n + 5^n - 5^n + 4^n \)
(c) \( 6^n + 5^n - 5^n - 4^n \)
(d) \( 6^n - 5^n - 4^n \)
Answer: (a) \( 6^n - 5^n - 5^n + 4^n \)

Question. Let N be the number of 4 digit numbers formed with at most two distinct digits. Then the last digit of N is
(a) 6
(b) 7
(c) 8
(d) 9
Answer: (a) 6

Question. If N is the least natural number which leaves remainders 2, 4, 6, 10 when divided by 3, 5, 7, 11 respectively, then the no. of divisors of N is
(a) 4
(b) 5
(c) 6
(d) 7
Answer: (a) 4

Question. Nine points lie in a plane forming a square as shown
.   .   .
.   .   .
.   .   .
If N is the number of triangles with positive area, having three of these points as vertices, then the last digit of N is
(a) 6
(b) 7
(c) 8
(d) 9
Answer: (a) 6

Question. Let n by any odd positive integer. The greatest integer k such that \( 2^k \) divides \( n^{12} - n^8 - n^4 + 1 \) is
(a) 9
(b) 10
(c) 11
(d) 12
Answer: (a) 9

Question. The number of ways of selecting two \( 1 \times 1 \) squares from a chess board such that they neither have a common vertex nor have a common side is N then last digit of N is
(a) 3
(b) 4
(c) 5
(d) 6
Answer: (d) 6

Question. \( A_1, A_2, A_3, A_4, A_5 \) lie on \( L_1 \). \( B_1, B_2, B_3, B_4, B_5 \) lie on \( L_2 \) and if \( L_1 \) is parallel to \( L_2 \). If these points are joined and lines are formed, the maximum number of points of intersection that will form between the lines is
(a) 80
(b) 100
(c) 120
(d) 240
Answer: (b) 100

Question. Let \( f : A \to A \) be an invertible function where \( A = \{1, 2, 3, 4, 5, 6\} \) The number of these functions in which at least three elements have self image is
(a) 40
(b) 56
(c) 16
(d) 3
Answer: (b) 56

Question. The number of ways of dividing 15 men and 15 women into 15 couples, each consisting of a man and a woman is
(a) 1240
(b) 1840
(c) 1820
(d) 2005
Answer: (a) 1240

Question. The number of words of four letters containg equal number of vowels and consonants, repetition allowed is
(a) \( 105^2 \)
(b) \( 210 \times 243 \)
(c) \( 105 \times 243 \)
(d) \( 150 \times 241 \)
Answer: (b) \( 210 \times 243 \)

Question. The number of ways of choosing 3 squares from a chess board so that they have exactly one common vertex
(a) 195
(b) 196
(c) 197
(d) 198
Answer: (b) 196