Question

A group consists of 9 men and 6 women. A team of 6 is to be selected. How many of possible selections will have at least 3 women?

Answer 1

Number of men = 9
Number of women = 6
Number of persons in the team = 6

Since the team of 6 is to include at least 3 women, therefore there will be 4 types of teams.
(i) 3 men and 3 women
(ii) 2 men and 4 women,
(iii) 1 man and 5 women and
(iv) all the 6 women and no man

(i) 3 men and 3 women:

The number of ways of forming the team = `""^9"C"_3 × ""^6"C"_3`

= `(9 × 8 × 7)/(3 × 2 × 1) × (6 × 5 × 4)/(3 × 2 × 1)`

= 84 × 20

= 1680 ways

(ii) 2 men and 4 women:

The number of ways of forming the team = `""^9"C"_2 × ""^6"C"_4`

= `(9 × 8)/(2 × 1) × (6 × 5 × 4 × 3)/(1 × 2 × 3 × 4)`

= 36 × 15

= 540 ways

(iii) 1 man and 5 women:

The number of ways of forming the team = `""^9"C"_1 × ""^6"C"_5`
= 9 × 6
= 54 

(iv) All the 6 women and no man:

The number of ways of forming the team = `""^9"C"_0 × ""^6"C"_6`
= 1 × 1
= 1 way

∴ The total number of ways of forming the team = 1680 + 540 + 54 + 1 = 2275

∴ 2275 teams can be formed if the team consists of at least 3 women.