NCERT Solutions Class 11th Maths Chapter – 7 Permutations and Combinations Exercise 7.4

NCERT Solutions Class 11th Maths Chapter – 7 Permutations and Combinations Exercise 7.4

NCERT Solutions Class 11th Maths Chapter – 7 Permutations and Combinations Exercise 7.4

?Chapter – 7?

✍Permutations and Combinations✍

?Exercise 7.4?

1. If ⁿC₈ = ⁿC₂, find ⁿC_2.

Solution:

2. Determine n if
(i) ²ⁿC_3:ⁿC₃ = 12: 1
(ii) ²ⁿC₃: ⁿC₃ = 11: 1

Solution:

Simplifying and computing
⇒ 4 × (2n – 1) = 12 × (n – 2)
⇒ 8n – 4 = 12n – 24
⇒ 12n – 8n = 24 – 4
⇒ 4n = 20
∴ n = 5

⇒ 11n – 8n = 22 – 4
⇒ 3n = 18
∴ n = 6

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

Solution:
Given 21 points on a circle
We know that we require two points on the circle to draw a chord
∴ Number of chords is are
⇒ ⁵C₃ × ⁴C₃ =

∴ Total number of chords can be drawn are 210

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

Solution:
Given 5 boys and 4 girls are in total
We can select 3 boys from 5 boys in ⁵C₃ ways
Similarly, we can select 3 boys from 54 girls in ⁴C₃ ways
∴ Number of ways a team of 3 boys and 3 girls can be selected is  ⁵C₃ × ⁴C₃

⇒ ⁵C₃ × ⁴C₃ = 10 × 4 = 40
∴ Number of ways a team of 3 boys and 3 girls can be selected is ⁵C₃ × ⁴C₃ = 40 ways

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.

Solution:
Given 6 red balls, 5 white balls and 5 blue balls
We can select 3 red balls from 6 red balls in ⁶C₃ ways
Similarly, we can select 3 white balls from 5 white balls in ⁵C₃ ways
Similarly, we can select 3 blue balls from 5 blue balls in ⁵C₃ ways
∴ Number of ways of selecting 9 balls is ⁶C₃ ×⁵C₃ × ⁵C₃

∴ 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

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

Solution:
Given a deck of 52 cards
There are 4 Ace cards in a deck of 52 cards.
According to question, we need to select 1 Ace card out the 4 Ace cards
∴ Number of ways to select 1 Ace from 4 Ace cards is ⁴C₁
⇒ More 4 cards are to be selected now from 48 cards (52 cards – 4 Ace cards)
∴ Number of ways to select 4 cards from 48 cards is  ⁴⁸C₄

∴ Number of 5 card combinations out of a deck of 52 cards if there is exactly one ace in each combination 778320.

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?

Solution:
Given 17 players in which only 5 players can bowl if each cricket team of 11 must include exactly 4 bowlers
There are 5 players how bowl, and we can require 4 bowlers in a team of 11
∴ Number of ways in which bowlers can be selected are: ⁵C₄
Now other players left are = 17 – 5(bowlers) = 12
Since we need 11 players in a team and already 4 bowlers are selected, we need to select 7 more players from 12.
∴ Number of ways we can select these players are: ¹²C₇
∴ Total number of combinations possible are: ⁵C₄ × ¹²C₇

∴ Number of ways we can select a team of 11 players where 4 players are bowlers from 17 players are 3960

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.

Solution:
Given a bag contains 5 black and 6 red balls
Number of ways we can select 2 black balls from 5 black balls are ⁵C₂
Number of ways we can select 3 red balls from 6 red balls are ⁶C₃
Number of ways 2 black and 3 red balls can be selected are ⁵C₂× ⁶C₃

∴ Number of ways in which 2 black and 3 red balls can be selected from 5 black and 6 red balls are 200

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?

Solution:
Given 9 courses are available and 2 specific courses are compulsory for every student
Here 2 courses are compulsory out of 9 courses, so a student need to select 5 – 2 = 3 courses
∴ Number of ways in which 3 ways can be selected from 9 – 2(compulsory courses) = 7 are ⁷C₃

∴ Number of ways a student selects 5 courses from 9 courses where 2 specific courses are compulsory are: 35