A club with 17 women and 21 men needs to form a committee of

A club with 17 women and 21 men needs to form a committee of size 7. How many committees are possible if a committee must have at least 3 women? And how many committees of seven different executive positions are possible? (for example, president, vice president, treasurer, etc) Please explain work. Thanks.

Solution

There are 17 women and 21 men

Committee size =7

a) 3 women must be included means

either 3w, 4m or 4w, 3m, or 5w,2m,or 6w, 1m or 7 w

Hence total no of ways for this selection =

(17C3)21C4+(17C4)21C5+(17C4)21C2+(17C6)21C1+(17C7)

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For 7 different positions, say position1, ii,iii,....vii.

For I position there are totally 38 choices. For ii 37, etc.

Hence no of position = 37P7