Class 11 Mathematics Combinations MCQs Set 06

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

Question. Number of squares of all dimensions of \( 5 \times 7 \) game board
(a) 70
(b) 75
(c) 80
(d) 85
Answer: (d) 85

Question. Maximum number of point of intersection made by 5 circles and 3 triangles
(a) 70
(b) 80
(c) 128
(d) 100
Answer: (c) 128

Question. If there are five periodes in each working day of a school, then the number of ways that you can arrange 3 subjects during the working day is
(a) 125
(b) 150
(c) 160
(d) 170
Answer: (b) 150

Question. 8 points are marked on the circumference of a circle at equal distances. Then the squares can be drawn by joining them?
(a) \( ^8P_4 \)
(b) \( ^8C_4 \)
(c) \( \frac{^8C_4}{2} \)
(d) 2
Answer: (d) 2

Question. The number of ways in which one or more balls can be selected out of 10 white, 9 greeen and 7 black balls is
(a) 892
(b) 881
(c) 891
(d) 879
Answer: (d) 879

Question. If \( s_n = \sum_{r=0}^n \frac{1}{^nC_r} \) and \( t_n = \sum_{r=0}^n \frac{r}{^nC_r} \), then \( \frac{t_n}{s_n} \)
(a) \( \frac{1}{2} n \)
(b) \( \frac{2n - 1}{2} \)
(c) \( n - 1 \)
(d) \( \frac{1}{2} n - 1 \)
Answer: (a) \( \frac{1}{2} n \)

Question. From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf so that the dictionary is always in the middle. Then the number of such arrangements is
(a) at least 500 but less than 750
(b) at least 750 but less than 1000
(c) at least 1000
(d) less than 500
Answer: (c) at least 1000

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
I: so as to include exactly 2 women is 210.
II: so as to include atleast 2 women is 301.
Which of the above statements is true?
(a) only I
(b) only II
(c) Both I and II
(d) Neither I nor II
Answer: (c) Both I and II
Hint:
I: \( ^{7}C_{3} \times ^{4}C_{2} = 210 \)
\( \implies \) true
II: \( ^{7}C_{3} \times ^{4}C_{2} + ^{7}C_{2} \times ^{4}C_{3} + ^{7}C_{1} \times ^{4}C_{4} = 301 \)
\( \implies \) true

Question. I. The no.of ways of dividing 15 different objects into 3 equal groups is \( \frac{15!}{5!5!5!} \)
II. The no.of ways in which 52 cards can be distributed among 4 persons equally is \( \frac{52!}{(13!)^{4} \cdot 4!} \)
Which of the above statement is correct?
(a) Only I
(b) Only II
(c) Both I and II
(d) Neither I nor II
Answer: (d) Neither I nor II
Hint:
I: \( \frac{15!}{(5!)^{3}3!} \)
\( \implies \) false
II: \( \frac{52!}{(13!)^{4}} \)
\( \implies \) false

Question. I : The no. of 3 digit numbers of the form xyz where x>y>z is 1.
II : The no.of 3 digit numbers of the form xyz. where x>y>z is \( ^{10}C_{3} 3! \)
Which of the above statement is correct?
(a) only I
(b) only II
(c) Both I and II
(d) Neither I nor II
Answer: (d) Neither I nor II
Hint:
I: Select 3 digits from 10 digits {0, 1, 2, ......., 9} in \( ^{10}C_{3} \) ways and arrange as can be given. This is done in 1 way
\( \implies \) \( ^{10}C_{3} \) (false)
II: \( ^{10}C_{3} \) is true

Question. In the shop there are five types of ice-creams available. A child buys six ice-creams.
Statement - 1 : The number of different ways the child can buy the six ice - creams is \( ^{10}C_{5} \)
Statement - 2 : The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging 6 A’s and 4 B’s in a row
(a) Statement-1 is true, Statement-2 is true and Statement - 2 is not a correct explanation for Statement - 1
(b) Statement-1 is true, Statement-2 is false
(c) Statement-1 is false, Statement-2 is true
(d) Statement-1 is true, Statement-2 is true and Statement-2 is a correct explanation for Statement-1
Answer: (c) Statement-1 is false, Statement-2 is true
Hint:
\( x_{1} + x_{2} + x_{3} + x_{4} + x_{5} = 6 \quad x_{k} \ge 0 \)
No. of solutions = \( ^{6+5-1}C_{5-1} = ^{10}C_{4} = \frac{10!}{6!.4!} \)

Question. Arrange the following values in ascending order.
A : no.of divisors of 24
B : no.of divisors of 12
C : no.of divisors of 72
D : no.of divisors of 120
(a) BACD
(b) DCAB
(c) BADC
(d) ABCD
Answer: (a) BACD
Hint:
A : \( 24 = 2^{3} \times 3^{1} \)
\( \implies \) \( 4 \times 2 = 8 \)
B : \( 12 = 2^{2} \times 3^{1} \)
\( \implies \) \( 3 \times 2 = 6 \)
C : \( 72 = 2^{3} \times 3^{2} \)
\( \implies \) \( 4 \times 3 = 12 \)
D : \( 120 = 2^{3} \times 3^{1} \times 5^{1} \)
\( \implies \) \( 4 \times 2 \times 2 = 16 \)
Ascending order BACD

Question. Arrange the following values in ascending order.
A : No.of diagonals of a polygon with 10 sides
B : No.of squares (exclusively squares) in a chess board
C : No.of ways in which 4 boys and six girls sit alternately in a row
D : No.of sides of the polygon in which no.of sides is equal to no.of diagonals.
(a) BADC
(b) DCAB
(c) CDBA
(d) CDAB
Answer: (d) CDAB
Hint:
A : \( \frac{10(7)}{2} = 35 \), B : \( \sum_{n=1}^{8} n^{2} = 204 \) C: 0 D : 5
Ascending order is CDAB

Question. Assertion(A): If a polygon has 35 diagonals then the number of the sides of the polygon is 15.
Reason(R): The number of diagonals of a polygon with ‘n’ sides is \( \frac{n(n-3)}{2} \)
(a) Both A and R are true and R is the correct explanation of A
(b) Both A and R are true and R is not correct explanation of R
(c) A is true but R is false
(d) A is false but R is true.
Answer: (d) A is false but R is true.
Hint:
\( \frac{n(n-3)}{2} = 35 \)
\( \implies \) \( n(n-3) = 10 \cdot 7 \)
\( \implies \) \( n = 10 \)
A is false & R is true

Question. Assertion (A): The no.of parallelograms in a chess board is 1296.
Reason(R): The no.of parallelograms when a set of ‘m’ parallel lines is intersected by another set of ‘n’ parallel lines is \( ^{m}C_{2} \cdot ^{n}C_{2} \)
(a) A is true and R is false
(b) A is false and R is true
(c) Both A and R are true and R is correct expalantion of A
(d) Both A and R are true and R is not correct explanation of A.
Answer: (c) Both A and R are true and R is correct expalantion of A
Hint:
A : \( ^{9}C_{2} \times ^{9}C_{2} = 1296 \) - true
R is true and correct explanation of A

Question. If given n points are on the circumference of a circle then observe the following Lists:
                        List = I                                     List - II
A. Number of straight lines               i. \( 2^{n} - 1 \)
B. Number of diagonals in an
    n-sided closed polygon                ii. \( ^{n}C_{4} \)
C. Number of triangles                     iii. \( ^{n}C_{2} - n \)
D. Number of quadrilaterals            iv. \( ^{n}C_{3} \)
                                                            v. \( ^{n}C_{2} \)
The correct Match for List - I from List - II is
(a) A-ii, B-i, C-iv, D-i
(b) A-v, B-iv, C-iii, D-i
(c) A-v, B-iii, C-iv, D-ii
(d) A-i, B-iii, C-ii, D-iv
Answer: (c) A-v, B-iii, C-iv, D-ii
Hint:
A : \( ^{n}C_{2} \), B : \( ^{n}C_{2} - n \), C : \( ^{n}C_{3} \), D : \( ^{n}C_{4} \)

Question. I: The total number of ways in which a selection(one or more) can be made out of \( p+q+r \) things of which p are all alike, q all alike, r all alike is \( (p+1)(q+1)(r+1)-1 \)
II : The number of permutations of ‘n’ things taken together when ‘p’ of the things are alike of one kind, ‘q’ of them alike of a second kind, ‘r’ of them alike of a third kind and the rest all different is \( \frac{n!}{p!q!r!3!} \)
Which of the above statement is true?
(a) only I
(b) only II
(c) Both I and II
(d) Neither I nor II
Answer: (a) only I
Hint:
I : true
II : \( \frac{n!}{p!q!r!} \)
\( \implies \) false

Question. Statement -1 : The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is \( ^{9}C_{3} \)
Statement-2 : The number of ways of choosing any 3 places from 9 different places is \( ^{9}C_{3} \)
(a) Statement-1 is true, statement-2 is false
(b) Statement-1 is false, Statement-2 is true.
(c) statement-1 is true, statement-2 is true, statement-2 is a correct explanation for statement1
(d) Statement-1 is true, Statement-2 is true, statement-2 is not a correct explanation for statement-1
Answer: (d) Statement-1 is true, Statement-2 is true, statement-2 is not a correct explanation for statement-1
Hint:
The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is \( ^{n-1}C_{r-1} = ^{10-1}C_{4-1} = ^{9}C_{3} \)
The number of ways of chooosing 3 places from 9 different places is \( ^{n}C_{r} = ^{9}C_{3} \).

Question. If a is the positive odd divisors of 216, b is the number of even divisors of 1600 and c is the number of odd divisors of 36000 then descending order of a, b, c is
(a) a,b,c
(b) b,c,a
(c) c,a,b
(d) a,c,b
Answer: (b) b,c,a
Hint:
\( 216 = 2^{3} \times 3^{3} \); No. of odd divisors = (3 + 1) = 4
\( 1600 = 2^{6} \times 5^{2} \); No. of even divisors=6(2+1)= 18
\( 36000 = 2^{5} \times 3^{2} \times 5^{3} \)
No. of odd divisors = (2 + 1) (3 + 1) = 12
Descending order is b, c, a

Question. Assertion (A): The number of positive divisors of \( 2^{5} 3^{6} 7^{3} \) is 168
Reason (R): The number of positive divisors of \( x^{n} y^{m} z^{r} \) ( here x, y and z are prime numbers) is \( (x+3)(y+4)(z-4) \)
(a) Both A and R are true and R is the correct explanation of A
(b) Both A and R are true and R is not correct explanation of R
(c) A is true but R is false
(d) A is false but R is true.
Answer: (c) A is true but R is false
Hint:
No. of positive divisors = \( 6 \times 7 \times 4 = 168 \) is true
R : \( (n+1)(m+1)(r+1) \)
\( \implies \) false

Question. Observe the following Lists            
            ​​​​​​​            ​​​​​​​            ​​​​​​​            ​​​​​​​List - I                                                                             ​​​​​​​             List - II
A. The number of ways of answering one or more of n questions                     (i) \( \frac{^{n}P_{r}}{2r} \)
B. The number of ways of answering one or more of n
questions when each question has an alternative is                                           (ii) \( 2^{n} - 1 \)
C. The number of circular permutations of n
different things taken r at a time is                                                                        (iii) \( \frac{^{n}P_{r}}{r} \)
D. The number of circular permutations of n things
taken r at a time in one direction is                                                                      ​​​​​​​  (iv) \( 3^{n} - 1 \)
                                                                                                                                   ​​​​​​​ (v) \( 2^{n} \)
The correct match is
(a) A-ii, B-iv, C-iii, D-i
(b) A-ii, B-iii, C-i, D-iv
(c) A-iii, B-ii, C-i, D-iv
(d) A-iv, B-iii, C-ii, D-i
Answer: (a) A-ii, B-iv, C-iii, D-i
Hint:
A : \( ^{n}C_{1} + \dots + ^{n}C_{n} = 2^{n} - 1 \)
B : \( 3^{n} - 1 \)
C : \( \frac{^{n}P_{r}}{r} \)
D : \( \frac{^{n}P_{r}}{2r} \)

PASSAGE : 1
Five balls are to be placed in three boxes. Each can hold all the five balls. The no. of ways of placing the balls so that no box remains empty

Question. If balls and boxes are all different is
(a) 50
(b) 100
(c) 125
(d) 150
Answer: (d) 150
Hint:
Number of ways \( (3^{5} - 3 \cdot 2^{5} + 3) = 150 \)

Question. If balls are identical but boxes are different is
(a) 2
(b) 6
(c) 4
(d) 8
Answer: (b) 6
Hint:
\( x_{1} + x_{2} + x_{3} = 5 \) ; \( x_{1}, x_{2}, x_{3} \ge 1 \)
Number of ways = \( ^{4}C_{2} = 6 \)

Question. If balls are different but boxes are identical is
(a) 25
(b) 15
(c) 10
(d) 35
Answer: (a) 25
Hint:
Possibilities are (3,1,1), (2,2,1). No. of ways = \( \frac{^{5}C_{3} \cdot ^{2}C_{1} \cdot ^{1}C_{1}}{2!} + \frac{^{5}C_{2} \cdot ^{3}C_{2} \cdot ^{1}C_{1}}{2!} = (10+15) = 25 \)

Question. If balls as well as boxes are identical is
(a) 1
(b) 2
(c) 25
(d) 150
Answer: (b) 2
Hint:
Only two ways (3,1,1), (2,2,1)