Permutation, Combination and Probability (UPSC) Part-II

Total Questions: 51

11. For a sports meet, a winners’ stand comprising three wooden blocks is in the following form : [2018-II]

There are six different colours available to choose from and each of the three wooden blocks is to be painted such that no two of them has the same colour. In how many different ways can the winners’ stand be painted?

(a) 120
(b) 81
(c) 66
(d) 36

Correct Answer: (a) 120

Solution:Required number of ways = 6! / 3! = 120

12. How many diagonals can be drawn by joining the vertices of an octagon? [2018-II]

(a) 20
(b) 24
(c) 28
(d) 64

Correct Answer: (a) 20

Solution:Number of diagonals in an octagon = ⁸C₂ - 8
= 8×7/2 - 8 = 20

13. While writing all the numbers from 700 to 1000, how many numbers occur in which the digit at hundred’s place is greater than the digit at ten’s place, and the digit at ten’s place is greater than the digit at unit’s place? [2018-II]

(a) 61
(b) 64
(c) 85
(d) 91

Correct Answer: (c) 85

Solution:Hundred Corresponding Corresponding

So, total numbers = 21 + 28 + 36 = 85

14. A bag contains 15 red balls and 20 black balls. Each ball is numbered either 1 or 2 or 3. 20% of the red balls are numbered 1 and 40% of them are numbered 3. Similarly, among the black balls, 45% are numbered 2 and 30% are numbered 3. A boy picks a ball at random. He wins if the ball is red and numbered 3 or if it is black and numbered 1 or 2. What are the chances of his winning? [2018-II]

(a) 1/2
(b) 4/7
(c) 5/9
(d) 12/13

Correct Answer: (b) 4/7

Solution:

Total number of balls 15 + 20 = 35
Total number of red balls numbered 3
= 15 × ( (100 - 20 - 40) / 100 ) = 15 × 40/100 = 6

Total number of black ball numbered 1 or 2
= 20 × ( (100 - 30) / 100 ) = 20 × 70/100 = 14

Required chances = (6 + 14)/35 = 4/7

15. Each face of a cube can be painted in black or white colours. In how many different ways can the cube be painted? [2019-II]

(a) 9
(b) 10
(c) 11
(d) 12

Correct Answer: (b) 10

Solution:

A cube has 6 symmetric faces.
Number of faces of a cube coloured by:

Black	White	Number of way
0	6	1
1	5	1
2	4	2 (two adjacent or two opposite faces are black)
3	3	2 (around a corner or not around a corner are either black or white)
4	2	2 (adjacent or opposite faces are white)
5	1	1
6	0	1

Hence, total number of ways = 1 + 1 + 2 + 2 + 2 + 1 + 1 = 10

16. How many triplets (x, y, z) satisfy the equation x + y + z = 6, where x, y and z are natural numbers? [2019-II]

(a) 4
(b) 5
(c) 9
(d) 10

Correct Answer: (d) 10

Solution:

Total Number of triplets = 10.

17. A printer numbers the pages of a book starting with 1 and uses 3089 digits in all. How many pages does the book have? [2019-II]

(a) 1040
(b) 1048
(c) 1049
(d) 1050

Correct Answer: (c) 1049

Solution:Pages till 9 uses 9 digits i.e.; 1 to 9.
Pages from 10 to 99 uses 2 digits each.
Number of pages = (99 - 10) + 1 = 90
Number of digits = 2 × 90 = 180.
Pages from 100 to 999 uses 3 digits each.
Number of pages = 999 - 100 + 1 = 900.
Number of digits = 3 × 900 = 2700.
Pages from 1000 to 9999 uses 4 digits each.
Remaining digits = 3089 - 2700 - 180 = 200.

Remaining number of pages = 200/4 = 50.

Hence, total number of pages = 999 + 50 = 1049.

18. Suppose you have sufficient amount of rupee currency in three denominations: ₹1, ₹10 and ₹50. In how many different ways can you pay a bill of ₹107? [2019-II]

(a) 16
(b) 17
(c) 18
(d) 19

Correct Answer: (c) 18

Solution:

Number of Denominations are shown in the table:

₹ 50	₹ 10	₹ 1	Number of ways
2	0	7	1
1	(0–5)	(7–57)	6
0	(0–10)	(7–107)	11

Total number of ways = 1 + 6 + 11 = 18.

19. The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of four parallel lines, is [2019-II]

(a) 18
(b) 24
(c) 32
(d) 36

Correct Answer: (d) 36

Solution:

Number of parallelograms that can be formed from 'm' parallel horizontal and 'n' parallel vertical lines
= ₄C₂ × ₄C₂

Here m = n = 4.

So, Number of prallelograms = ₄C₂ × ₄C₂ = 6 × 6 = 36.

20. One page is torn from a booklet whose pages are numbered in the usual manner starting from the first page as 1. The sum of the numbers on the remaining pages is 195. The torn page contains which of the following numbers? [2020-II]

(a) 5, 6
(b) 7, 8
(c) 9, 10
(d) 11, 12

Correct Answer: (b) 7, 8

Solution:

Let, there are n number of pages in the book.
Sum of page numbers = n(n+1)/2

∴ n(n+1) = 2 × (Sum of page numbers)
Here given, sum of page numbers = 195.
∴ n(n+1) = 195 × 2 + Z

where, Z = 2 × (Sum of two continuous page number that is torn out)

For, n = 20, 20 × 21 = 390 + Z
⇒ Z = 420 – 390 = 30

⇒ Z/2 = 15

Here, Z/2 = 7 + 8

Hence, torn page number is 7, 8.

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Permutation, Combination and Probability (UPSC) Part-II

11. For a sports meet, a winners’ stand comprising three wooden blocks is in the following form : [2018-II]

12. How many diagonals can be drawn by joining the vertices of an octagon? [2018-II]

13. While writing all the numbers from 700 to 1000, how many numbers occur in which the digit at hundred’s place is greater than the digit at ten’s place, and the digit at ten’s place is greater than the digit at unit’s place? [2018-II]

15. Each face of a cube can be painted in black or white colours. In how many different ways can the cube be painted? [2019-II]

16. How many triplets (x, y, z) satisfy the equation x + y + z = 6, where x, y and z are natural numbers? [2019-II]

17. A printer numbers the pages of a book starting with 1 and uses 3089 digits in all. How many pages does the book have? [2019-II]

18. Suppose you have sufficient amount of rupee currency in three denominations: ₹1, ₹10 and ₹50. In how many different ways can you pay a bill of ₹107? [2019-II]

19. The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of four parallel lines, is [2019-II]

20. One page is torn from a booklet whose pages are numbered in the usual manner starting from the first page as 1. The sum of the numbers on the remaining pages is 195. The torn page contains which of the following numbers? [2020-II]

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