NCERT Solutions for Exercise 6.4, Class 11, Maths

Exercise 6.4
Permutations and combinations
Question and Answers
Class 11 – Maths

Class	Class 11
Subject	Mathematics
Chapter Name	Permutations and Combinations
Chapter No.	Chapter 6
Exercise	Exercise 6.4
Category	Class 11 Maths NCERT Solutions

Question 1 If ⁿC₈ = ⁿC₂, find ⁿC₂.

Answer ⁿC_a = ⁿC_b

⇒ a = b or n = a+ b

n = 8 + 2 =10

Therefore

Question 2 Determine n , if

(i) ²ⁿC₃ = ⁿC₃ = 12 : 1 

⇒ 2n – 1 = 3(n – 2) 
⇒ 2n – 1 = 3n – 6 
⇒ 3n – 2n = -1 + 6 
⇒ n = 5

(ii) ²ⁿC₃ = ⁿC₃ = 11 : 1 

⇒ 4(2n – 1) = 11(n – 2)
⇒ 8n – 4 = 11n – 22 
⇒ 11n – 8n = -4 + 22 
⇒ 3n = 18 
⇒ n = 6

Question 3  How many chords can be drawn through 21 points on a circle?

Answer For drawing one chord a circle, only 2 points are required.

To know the number of chords that can be drawn through the given 21 points on a circle, the number of combinations have to be counted.Therefore, the chords can be drawn through 21 points taken 2 as equal to each chord.

Thus, required number of chords =

⇒ 210

Question 4 In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls?

Answer  A team of 3 boys and 3 girls is to be selected from 5 boys and 4 girls.
3 boys can be selected from 5 boys in  ⁵C₃  ways.
3 girls can be selected from 4 girls in ⁴C₃  ways.
Number of ways in which a team of 3 boys and  3 girls can be selected =

⇒40

Question 5 Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if each selection consists of 3 balls of each colour.

Answer There are a total of 6 red balls, 5 white balls, and 5 blue balls.
9 balls have to be selected in such a way that each selection consists of 3 balls of each colour.
Here,
3 balls can be selected from 6 red balls in ⁶C₃ ways.
3 balls can be selected from 5 white balls in ⁵C₃ ways.
3 balls can be selected from 5 blue balls in ⁵C₃ ways.

Required number of ways of selecting the 9 balls will be –

∴ The number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if each selection consists of 3 balls of each colour is ⁶C₃ ×⁵C₃ × ⁵C₃ = 2000

Question 6 Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly one ace in each combination.

Answer We have a deck of 52 cards, which contains 4 aces. When we make a combination, 5 cards should be made in such a manner that there is exactly one ace.

Then, one ace can be selected in ⁴C₁ ways and the remaining 4 cards can be selected out of the 48 cards in ⁴⁸C₄ ways.

The required number of 5 card combinations will be –

Question  7 In how many ways can one select a cricket team of eleven from 17 players in which only 5 players can bowl if each cricket team of 11 must include exactly 4 bowlers?

Answer The number of players out of which we have to select is 17 players and only 5  players from them are bowlers.

A cricket team of 11 players is to be selected in such a way that there are exactly 4 bowlers.

4 bowlers can be selected in ⁵C₄ ways and the remaining 7 players can be selected out of the 12 players in  ¹²C₇ ways.

Required number of ways of selecting cricket team –

Question  8 A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and 3 red balls can be selected.

Answer There are 5 black and 6 red balls in the bag.

2 black balls can be selected out of 5 black balls in ⁵C₂ ways and 3 red balls can be selected out of 6 red balls in ⁶C₃ ways.

Required number of ways of selecting 2 black and 3 red balls  –

Question 9 In how many ways can a student choose a programme of 5 courses if 9 courses are available and 2 specific courses are compulsory for every student?

Answer There are 9 courses available out of which, 2 specific courses are compulsory for every student.

Therefore, every student has to choose 3 courses out of the remaining 7 courses. This can be chosen in ⁷C₃ ways.

Thus, required number of ways of choosing the programme –