Question: 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.

1. 12
2. 21
3. 36
4. 42
5. 46

Correct Answer: D
Solution and Explanation:
Approach Solution 1:

One married couple only:
Select one married couple out of 4 in 4C1 ways.
Select one male for the other team in 3 ways and one non-wife female in 2 ways.
Number of games with only one married couple = 4*3*2 = 24

Both married couples
Select 2 married couples in 4C2 = 6 ways

Number of games in which at least there will be one couple = 24+6 = 30

Total number of games = (4*4 * 3*3)/2 = 72
Select team 1 in 4*4 ways and team 2 in 3*3 ways. Divide by 2 because you don't want to arrange the teams in team 1 and team 2. They are just 2 teams.
So in 72 - 30 = 42 games, there will be no married couple.

Approach Solution 2:

Married couples: MF MF MF MF
                              ab cd ef gh
Possible teams: ad cd eb gb
                            af cf ed gd
                            ah ch eh gf

Now, team ad can play only with: cb, cg, ch, eb, eh, gb, gf, i.e. 7
The same will apply with all teams.
So no. of total match = 12 × 7 = 84
Since every match includes 2 teams,
so the no. of matches = 84/2 = 42

Approach Solution 3:
After we have chosen the pair of husbands, A and B:
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 = 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 = 7 possibilities to choose their partners for the game.
Total number of possibilities 6 * 7 = 42.

“A mixed doubles tennis game is to be played between two teams.”- is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

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