A mixed doubles tennis game is to be played between two teams. There are four married couples. No team is to consist of a husband and his wife. What is the maximum number of games that can be played.  Select the correct option
A) 12  B) 21  C) 36  D) 42  E) 46
Let's denote the four couples by (A,a), (B,b), (C,c), and (D,d) where A,B,C,D are the husbands and a,b,c,d are the wives.   For a games, let's choose first the husbands. We have 4C2=4*3/2= 
<gadget id="calculator">4*3/2</gadget>
<output>6</output>
6 possibilities. Now we know that the final number of games will be a multiple of 6, so we are down to two choices, C and D.   Once husbands were chosen, say A and B, let's count how many possibilities we have to choose their partners under the given restrictions.   We can choose a and b and we have to pair them as (A,b) and (B,a) - 1 possibility.   We can choose one of the wives, a or b, but we have to pair her with the other husband and in addition, we have to choose another partner for the second husband.   This we can do in 2*2= 
<gadget id="calculator">2*2</gadget>
<output>4</output>
4 ways, as there are two possibilities to choose from a and b, then we have 2 possibilities to choose the other wife, c or d - 4 possibilities   Finally, we can choose the other two wives, c and d, and we have two possibilities to team them up with the men, (A,c), (B,d) or (A,d), (B,c) - 2 possibilities.   In conclusion, for every pair of husbands, we have 1 + 4 + 2= 
<gadget id="calculator">1 + 4 + 2</gadget>
<output>7</output>
7 possibilities to choose their partners for the game.   Total number of possibilities 6 * 7= 
<gadget id="calculator">6 * 7</gadget>
<output>42</output>
42.   Answer 42.
<result>D</result>
A mixed doubles tennis game is to be played between two teams. There are four married couples. No team is to consist of a husband and his wife. What is the maximum number of games that can be played.