2012
Problem - 227
At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and must also contain two professors from each of the three departments. Find the number of possible committees that can be formed subject to these requirements.
There are two cases to consider:
- One man and one woman chosen from each of these three departments. There are $\left(2\times 2\right)^3 = 64$ ways.
- Two men from one department, two women from another department, and one man one woman from the third department. There are $C_3^1$ ways to choose the first department, $C_2^1$ ways to chose the second department, and $2\times 2=4$ ways to pick one man and one woman from the third department. Hence, the total count in this case is $C_3^1\times C_2^1 \times 4=24$..
Therefore the answer is $64+24=\boxed{88}$.