A mixed doubles tennis game is to be played between two teams (each team consists of one male and one female). 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?

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Let the husband and wife couples be:

Alan and Alice

Barry and Barbara

Charles and Charlotte

David and Diane

Possible 12 teams are:

  1. Alan and Barbara
  2. Alan and Charlotte
  3. Alan and Diane
  4. Barry and Alice
  5. Barry and Charlotte
  6. Barry and Diane
  7. Charles and Alice
  8. Charles and Barbara
  9. Charles and Diane
  10. David and Alice
  11. David and Barbara
  12. David and Charlotte.

Now we create opposing teams by selecting a team, say team 1. Then we eliminate all teams containing the same players, that is, all teams containing Alan or Barbara. That eliminates teams 2, 3, 8, 11, leaving 7 eligible teams as opponents. Each team, in fact, will have 7 eligible opposing teams and there are 12 teams, so there can be 12×7=84 games. But, wait a minute! Let’s take just one game, say team 1 against team 7. When we get to looking at the games team 7 can play we will include team 7 against team 1. Unless, the position of the players in the tennis court is relevant, team 1 against team 7 is the same as team 7 against team 1. So we have to halve the number of games, making it 84/2=42 games.

The fixtures are shown above. X means there can be no game. The blank cells represent valid games, 42 in all.

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