Question

A committee of 7 peoples has to be formed from 8 men and 4 women. In how many ways can this be done when the committee consists of at least 3 women?

Answer 1

Number of men = 8

Number of women = 4

Number of peoples in the committee = 7

At least 3 women

The 7 members committee must contain at least 3 women

∴ We have the following possibilities

(i) 4 women + 3 men

(ii) 3 women + 4 men

Case (i): 4 women + 3 men

The number ways of selecting 4 women .from

4 women is = 4C₄ = 1 way

The number of ways of selecting 3 men from 8 men = 8C₃ 

= `(8!)/(3! xx (8 - 3)!)`

= `(8!)/(3! xx 5!)`

= `(8 xx 7 xx 6 xx 5!)/(3!xx 5!)`

= `(8 xx 7 xx 6)/(3 xx 2 xx 1)`

= 8 × 7

= 56

∴ The total number of ways = 1 × 56 = 56

Case (ii): 3 women + 4 men

The number of ways of selecting 3 women from 4 women is = 4C₃

The number of ways of selecting 4 men from 8 men is = 8C₄

∴ The total number of ways = 4C₃ × 8C₄ 

= `(4!)/(3! xx (4 - 3)!) xx (8!)/(4!(8 - 4)!)`

= `(4!)/(3! xx 1!) xx (8!)/(4! xx 4!)`

= `(4 xx 3!)/(3!) xx (8 xx 7 xx 6 xx 5 xx 4!)/(4! xx 4!)`

= `(4 xx 8 xx 7 xx 6 xx 5)/(4!)`

= `(4 xx 8 xx 7 xx 6 xx 5)/(4 xx 3 xx 2 xx 1)`

= 8 × 7 × 5 = 280

∴ The required number of ways of forming the committee = 56 + 280 = 336