M10-25

Official Solution:

A committee of three students has to be formed from five candidates: Jane, Joan, Paul, Stuart, and Jessica. If Paul and Stuart refuse to be on the committee together, and Jane refuses to be on the committee without Paul, how many committees can be formed?

A. 3
B. 4
C. 5
D. 6
E. 8


First, we note that a committee cannot be formed if neither Paul nor Stuart is on the committee. This is because, without Paul, Jane also refuses to be on the committee, leaving only two other candidates (Joan and Jessica), which is not enough to form a three-member committee.

Now, let's consider the committees that include Paul. Since Stuart refuses to be on the committee with Paul, we must choose two other members from the remaining candidates: Jane, Joan, and Jessica. This gives us \(C^2_3=3\) possible committees.

Similarly, we can count the committees that include Stuart but not Paul. Since Jane refuses to be on the committee without Paul, we must choose two other members from the remaining candidates: Joan and Jessica. This gives us \(C^2_2=1\) possible committee.

Thus, the total number of possible committees is \(3+1=4\).


Answer: B