a computer programming team has 15 people that contain 9 wom

a computer programming team has 15 people that contain 9 women and 6 men.

(a) how many groups of 10 can be chosen that contain 7 women and 3 men?

(b) how many groups of 10 can be chosen that contain at least 5 men?

Solution

number of people = 15

number of women = 9

number of men = 6

a)

we have to choose 7 women out of 9

= 9C₇ = 36 ways

3men out of 6

= 6C₃ = 20 ways

so total

groups of 10 can be chosen that contain 7 women and 3 men can be formed in

20 * 36 = 720 ways

b)

atleast 5 men means minimum 5 and a maximum of 6 men is possible

So

if there are 5 men and 5 women = 6C_{5 *} 9C₅ = 6 * 126 = 756 ways

if there are 6 men and 4 women = 6C_{6 *} 9C₄ = 1 * 126 = 126 ways

So total

groups of 10 can be chosen that contain at least 5 men are

756 ways + 126 ways = 882 ways