A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of:
(i) exactly 3 girls?   (ii) atleast 3 girls?   (iii) atmost 3 girls?

Solution:

We know that the number of ways of selecting r different things from n different things is a combination and is calculated using the formula ⁿCᵣ = n! / [r!(n−r)!]. Using this,

(i) No. of ways of to selecting 3 girls from 4 girls and 4 boys from 9 boys to form a committee of 7

= ⁴C₃ × ⁹C₄

= (4!) / [3! (4-3)!] × (9!) / [4! (9-4)!]  (Using nCr formula)

= 4 × 126

= 504.

(ii) atleast 3 girls 

This means that there can be either 3 or 4 girls in the committee of 7.

• No. of ways of to selecting 3 girls from 4 and 4 boys from 9 to form a committee of 7 = ⁴C₃ × ⁹C₄ = 4 × 126 = 504
• No. of ways of to selecting 4 girls from 4 and 3 boys from 9 to form a committee of 7 = ⁴C₄ × ⁹C₃ = 1 × 84 = 84

The total no. of ways = 504 + 84 = 588.

(iii) atmost 3 girls 

This means that there can be 1 or 2 or 3 girls in the committee of 7.

• No. of ways of to selecting 1 girl from 4 and 6 boys from 9 to form a committee of 7 = ⁴C₁ × ⁹C₆ = 4 × 84 = 336
• No. of ways of to selecting 2 girls from 4 and 5 boys from 9 to form a committee of 7 = ⁴C₂ × ⁹C₅ = 6 × 126 = 756
• No. of ways of to selecting 3 girls from 4 and 4 boys from 9 to form a committee of 7 = ⁴C₃ × ⁹C₄ = 4 × 126 = 504

The total no. of ways = 336 + 756 + 504 = 1596

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NCERT Solutions Class 11 Maths Chapter 7 Exercise ME Question 3

A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of: (i) exactly 3 girls? (ii) atleast 3 girls? (iii) atmost 3 girls?

Summary:

(i) The no. of ways of forming a committee with exactly 3 girls is 504.

(ii) The no. of ways of forming a committee with atleast 3 girls is 84

(iii) The no. of ways of forming a committee with atmost 3 girls is 1596