Exercise 4.5: Choose the correct or the most suitable answer

CHAPTER : Combinatorics and Mathematical Induction 

Choose the correct or the most suitable answer 

1. The sum of the digits at the 10^th place of all numbers formed with the help of 2, 4, 5, 7 taken all at a time is

(1) 432

(2) 108

(3) 36

(4) 18

Solution

2. In an examination there are three multiple choice questions and each question has 5 choices .

Number of ways in which a student can fail to get all answer correct is

(1) 125

(2) 124

(3) 64

(4) 63

Solution

3. The number of ways in which the following prize be given to a class of 30 boys first and second in mathematics, first and second in physics, first in chemistry and first in English is

(1) 30⁴ x 29²

(2) 30³ x 29³

(3) 30² x 29⁴

(4) 30 x 29⁵.

Solution

4. The number of 5 digit numbers all digits of which are odd is

(1) 25

(2) 5⁵

(3) 5⁶

(4) 625.

Solution

5. In 3 fingers, the number of ways four rings can be worn is ………..ways.

(1) 4³ - 1

(2) 3⁴

(3) 68

(4) 64

Solution

4 rings can be worn in 3 fingers in 34 ways

6. If ⁽ⁿ⁺⁵⁾P_(n+1) = ( 11(n-1) /2 )⁽ⁿ⁺³⁾P_n, then the value of n are

(1) 7 and 11

(2) 6 and 7

(3) 2 and 11

(4) 2 and 6.

Solution

7. The product of r consecutive positive integers is divisible by

(1) r!

(2) (r - 1)!

(3) (r + 1)!

(4) r^r.

Solution

Product of r consecutive positive integers is divisible by r! (theorem)

8. The number of five digit telephone numbers having at least one of their digits repeated is

(1) 90000

(2) 10000

(3) 30240

(4) 69760.

Solution

9. If a²-^aC₂ = ^a2-^a C₄ then the value of ’a’ is

(1) 2

(2) 3

(3) 4

(4) 5

Solution

10. There are 10 points in a plane and 4 of them are collinear. The number of straight lines joining any two points is

(1) 45

(2) 40

(3) 39

(4) 38.

Solution

11. The number of ways in which a host lady invite 8 people for a party of 8 out of 12 people of whom two do not want to attend the party together is

(1) 2 x¹¹C₇ +¹⁰C₈

(2) ¹¹C₇ +¹⁰C₈

(3) ¹²C₈ -¹⁰C₆

(4) ¹⁰C₆ + 2!.

Solution

12. The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines.

(1) 6

(2) 9

(3) 12

(4) 18

Solution

13. Everybody in a room shakes hands with everybody else. The total number of shake hands is 66. The number of persons in the room is            

(1) 11

(2) 12

(3) 10

(4) 6

Solution

14. Number of sides of a polygon having 44 diagonals is

(1) 4

(2) 4!

(3) 11

(4) 22

Solution

15. If 10 lines are drawn in a plane such that no two of them are parallel and no three are concurrent, then the total number of points of intersection are

(1) 45

(2) 40

(3)10!

(4) 2¹⁰

Solution

16. In a plane there are 10 points are there out of which 4 points are collinear, then the number of triangles formed is

(1) 110

(2) ¹⁰C₃

(3) 120

(4) 116

Solution

17. In ²ⁿC₃ :ⁿ C₃ = 11 : 1 then n is

(1) 5

(2) 6

(3)11

(4)7

Solution

18. ^(n-1)C_r +^(n-1) C_(r-1) is

(1) ⁽ⁿ⁺¹⁾C_r

(2) ^(n-1)C_r

(3) ⁿC_r

(4)ⁿC_r-1.

Solution

19. The number of ways of choosing 5 cards out of a deck of 52 cards which include at least one king is

(1) ⁵²C₅

(2) ⁴⁸C₅

(3)⁵²C₅ +⁴⁸ C₅

(4)⁵²C₅ -⁴⁸ C₅.

Solution

20. The number of rectangles that a chessboard has………..

(1) 81

(2) 9⁹

(3)1296

(4) 6561

Solution

21. The number of 10 digit number that can be written by using the digits 2 and 3 is

(1) ¹⁰C₂ + ⁹C₂

(2) 2¹⁰

(3)2¹⁰ - 2

(4) 10!

Solution

22. If P_r stands for ^rP_r then the sum of the series 1 + P₁ + 2P₂ + 3P₃ +……….+ nP_n is

(1) P_n+1

(2) P_n+1 - 1

(3) P_n-1 + 1

(4)⁽ⁿ⁺¹⁾P_(n-1)

Solution

23. The product of first n odd natural numbers equals

(1) ²ⁿC_n x ⁿP_n

(2) (1/2 )ⁿ x²ⁿC_n xn P_n

(3) (1/4 )ⁿ x ²ⁿC_n x ²ⁿ P_n

(4)ⁿC_n x ⁿP_n

Solution

24. If ⁿC₄, ⁿC₅, ⁿC₆ are in AP the value of n can be

(1) 14

(2) 11

(3)9

(4)5

Solution

25. 1 + 3 + 5 + 7 +……….+ 17 is equal to

(1) 101

(2) 81

(3) 71

(4) 61

Solution