UPSC Prelims 2006 Q102: Science & Technology – Quantitative Aptitude
A mixed doubles tennis game is to be played between two teams (each team consists of
one male and one female). There are 4 married couples. No team is to consist of a husband and
his wife. What is the maximum number of games that can be played?
- (a) 12
- (b) 21
- (c) 36
- (d) 42
Answer: (d) 42
Explanation
Mixed doubles games from four married couples
There are 4 men and 4 women. A team is one man and one woman who are not married to each other.
| Step | Calculation |
|---|---|
| Possible teams | 4 x 4 - 4 married pairs = 12 |
| All pairs of teams | C(12, 2) = 66 |
| Pairs sharing a man | 4 men x C(3, 2) = 12 |
| Pairs sharing a woman | 4 women x C(3, 2) = 12 |
| Valid games (no common player) | 66 - 12 - 12 = 42 |
The maximum number of games that can be played is 42.
Answer: (d) 42.
Why each option is right or wrong
- (a) Incorrect. 12 is only the number of possible teams, not the games.
- (b) Incorrect. 21 undercounts the valid disjoint pairs of teams.
- (c) Incorrect. 36 does not result from removing pairs that share a player.
- (d) CORRECT. 66 - 12 - 12 = 42 games between two disjoint teams.
Key point
There are 12 valid teams; of 66 pairs of teams, 24 share a player, leaving 42 games.
Remember
12 teams, C(12,2) = 66, minus 24 overlapping pairs = 42 games.
Common mistakes
- Counting games without removing team pairs that share a player.
- Forgetting to exclude husband-wife teams.
- Treating a game as an ordered pair of teams.
Exam tip
Count all pairs of teams, then subtract the pairs that share a man or a woman.
Sources
- NCERT Class XI Mathematics — Permutations and Combinations
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Permutations and Combinations)