Bunuel wrote:
The number of ways of choosing a committee of two women and three men from five women and six men, if Mr. A refuses to serve on the committee if Mr. B is a member and Mr. B can only serve, if Miss C is the member of the committee is
(A) 60
(B) 84
(C) 124
(D) 136
(E) 144
Solution:
There are 3 cases to consider: 1) Mr. A is on the committee, but not Mr. B, 2) Mr. B is on the committee, but not Mr. A, and 3) neither Mr. A nor Mr. B is on the committee. Let’s now analyze each case:
Case 1: Mr. A is on the committee, but not Mr. B.
If Mr. A is on the committee, then we only need to choose 2 more men from the 4 remaining men since Mr. B can’t be on the committee. Since we also need to choose 2 women from the 5 women, the number of ways this can be done is 4C2 x 5C2 = 6 x 10 = 60.
Case 2: Mr. B is on the committee, but not Mr. A. (and Miss C is on the committee)
If Mr. B is on the committee, then we only need to choose 2 more men from the 4 remaining men since Mr. A refuses to serve with Mr. B. Furthermore, since Miss C also needs to be on the committee, there is 1 more woman to choose from the 4 remaining women; the number of ways this can be done is 4C2 x 4C1 = 6 x 4 = 24.
Case 3: Neither Mr. A nor Mr. B is on the committee.
If neither Mr. A nor Mr. B is on the committee, then we need to choose 3 men from the remaining 4 men. Since we also need to choose 2 women from the 5 women, the number of ways this can be done is 4C3 x 5C2 = 4 x 10 = 40.
Therefore, the total number of ways this can be done is 60 + 24 + 40 = 124.
Answer: C