Question

In how many ways can a committee of 4 be made out of 5 men and 3 women containing at least one woman?
1. 65
2. 70
3. 75
4. 60

Answer 1

Correct Answer - Option 1 : 65

Formula used:

• \(^nC_r=\frac{n!}{r!(n-r)!}\)
• ⁿC_n = 1
• ⁿC_r = ⁿC_n-r

 

Calculation:

Given that,

There are 5 men and 3 women. 

Number of ways in which committee form containing at least 1 women

3 men & 1 women + 2 men & 2 women + 1 men & 3 women

= ⁵C₃ × ³C₁ + ⁵C₂ × ³C₂ + ⁵C₁ × ³C₃

= 2× 5C3 × 3C1 + 5C1 × 3C3   (∵nCr = nCn-r)

= \(2\times\frac{5!}{3!×(5-3)!}×3+5\times1\)     (∵ nCn = 1)

= \(2\times\frac{5\times4\times3 \times2!}{3!×(2)!}×3+5 \)

= 60 + 5

= 65