Bunuel wrote:
A committee of 12 is to be formed from 9 women and 8 men. In how many ways this can be done if at least five women have to be included in a committee?
A. 1008
B. 2702
C. 6062
D. 7702
E. 8702
Solution:Since the committee must have at least 5 women, we can have the following cases:
1) If 5 women are on the committee, then 7 men must also be on the committee, and hence there are 9C5 x 8C7 = 126 x 8 = 1,008 ways.
2) If 6 women are on the committee, then 6 men must also be on the committee, and hence there are 9C6 x 8C6 = 84 x 28 = 2,352 ways.
3) If 7 women are on the committee, then 5 men must also be on the committee, and hence there are 9C7 x 8C5 = 36 x 56 = 2,016 ways.
4) If 8 women are on the committee, then 4 men must also be on the committee, and hence there are 9C8 x 8C4 = 9 x 70 = 630 ways.
5) If all 9 women are on the committee, then 3 men must also be on the committee, and hence there are 9C9 x 8C3 = 1 x 56 = 56 ways.
Therefore, there are a total of 1,008 + 2,352 + 2,016 + 630 + 56 = 6,062 ways.
Alternate Solution:Since there are only 8 men, the smallest number of women on the committee is 4. We have the following equality:
Total number of committees = The number of committees with exactly 4 women + The number of committees with at least 5 women
Thus, we can find the number of committees with at least 5 women simply by subtracting the number of committees with exactly 4 women from the total number of committees.
Since there are 9 + 8 = 17 men and women, there are 17C12 = 17!/(12!*5!) = (17 x 16 x 15 x 14 x 13)/(5 x 4 x 3 x 2) = 17 x 14 x 13 x 2 = 6,188 committees in total.
The number of committees with exactly 4 women is 9C4 * 8C8 = 9C4 * 1 = 9!/(4!*5!) = (9 x 8 x 7 x 6)/(4 x 3 x 2) = 9 x 7 x 2 = 126.
Therefore, there are 6,188 - 126 = 6,062 committees with at least 5 women.
Answer: C _________________
See why Target Test Prep is the top rated GMAT course on GMAT Club.
Read Our Reviews