A club committee of five is to be formed from six women and

A club committee of five is to be formed from six women and four men. The six women include the president and the treasurer. Evaluate the number of ways the committee can be chosen if it must consist of three women and two men, there must be at least three women on it, the president and secretary cannot both serve on the same committee, a particular two sisters will only serve on a committee if they are both on it.

Solution

No of women 6

No of men 4

a) 3 w , 2 m

No of ways = 6C3 (4C2) = 120 ways

b) 3w, 2 m or 4 w, 1 m or 5 w

No of ways = 6C3 (4C2)+ 6C4 (4C1)+6C5

= 120+60+6

=186

c) one president and one treasurer from 6 women and remaining 3 from any of the remaining 7

6C2 * 7C3 = 15(35) = 525

d) 2 sisters can be taken as a single unit and the committee consists of 5 units excluding sisters or 4 units including sister.

8C5 + 7C5 = 84+21

=105