4GMAT_Mumbai wrote:Hi ... Interesting question ... Thanks.
Let us do this in two steps: (1) Selection of 4 people and (2) Arrangement.
1) Selection of 4 people:
2 men out of 5 men can be chosen in 5C2 ways (i.e., 10)
2 women out of 3 women (excluding the spouses of men already chosen) can be chosen in 3C2 ways (i.e., 3)
Hence, selection of 4 people can be done in 10 * 3 = 30 ways.
2) Arrangement of these 4 people:
Two men (A, B) and two women (C,D) can form teams as:
A,C on one side with B,D on another
A,D on one side with B,C on another.
Thus, there are only 2 ways of arrangement for every group of 4 people.
Hence, final answer = selection ways * arrangement ways = 30 times 4 = 120 ways.
This is a quarter of what kvcpk has come up with. Look forward to know the right answer and if there are flaws in this thinking.
Thanks.
This solution looks right to me until the very final calculation - you found there were 30 ways to select the people, and 2 ways to arrange them, so the multiplication at the end ought to be 30*2 = 60, and not 30*4 = 120 (I think you just plugged in the wrong number).
I like kvcpk's approach, but it overcounts the number of possibilities. First, the order of the players within each team is not important -- if we pick Alan first, then Maria, we get the same team as if we had picked Maria first and then Alan. In kvcpk's solution, these would be counted as though they were different, but they aren't, so we need to divide by 2 twice, once for each team. Further, the order of the *teams* themselves doesn't matter (if Alan+Maria play against Kumar+Elena, we have the same tennis match as when Kumar+Elena play against Alan+Maria). So we need to divide by 2 yet again. That's why kvcpk's answer is 8 times too large, but otherwise it's a nice way to look at the problem.
You could modify the approach to get the right answer. Rather than pick the first player from all 10 people, instead pick first the man, then the woman, for each team. You would have 5 choices for the man on the 'first' team, and 4 choices for the woman, then 3 choices for the man on the 'second' team and 2 choices for the woman, for 5*4*3*2 = 120 possible choices of teams. Here we have labeled the teams - we called one team the 'first' team and one the 'second', so we've arrived at the answer *if* the order of the teams is important. If the order of the teams is irrelevant, as I assume is the intended interpretation of the question, then we've double-counted, so we need to divide by 2, and the answer is 60.
I'd add two things: you would *never* need to know what a 'mixed doubles game' is on the GMAT (they aren't testing how much you know about tennis!), and I haven't seen any real GMAT counting questions that are quite this intricate, where you need to determine whether order matters at so many different stages of the solution, so I wouldn't be too concerned about studying this problem in detail.
