UPSC Prelims 2006 Q60: Science & Technology – Quantitative Aptitude
In a tournament each of the participants was to play one match against each of the other
participants. 3 players fell ill after each of them had played three matches and had to leave the
tournament. What was the total number of participants at the beginning, if the total number of
matches played was 75 ?
- (a) 8
- (b) 10
- (c) 12
- (d) 15
Answer: (d) 15
Explanation
Total participants in a round-robin tournament
Every participant plays one match against each other participant. Three players left after playing three matches each.
| Step | Calculation |
|---|---|
| Matches played by the 3 players who fell ill | 3 x 3 = 9 |
| Matches among the remaining players | Total 75 - 9 = 66 |
| Remaining players r | r(r - 1)/2 = 66, so r = 12 |
| Participants at the beginning | 12 + 3 = 15 |
So the tournament began with 15 participants.
Answer: (d) 15.
Why each option is right or wrong
- (a) Incorrect. 8 participants would give far fewer than 75 matches.
- (b) Incorrect. 10 participants give at most 45 matches in a full round robin.
- (c) Incorrect. 12 is the number remaining after three left, not the starting number.
- (d) CORRECT. 12 remaining players give 66 matches, and 9 matches played by the three who left make 75; 12 + 3 = 15.
Key point
The three who left played 9 matches, the remaining 12 played 66 among themselves, so 12 + 3 = 15 participants started.
Remember
75 matches: 9 + 66; 12 remaining + 3 who left = 15 participants.
Common mistakes
- Using n(n-1)/2 = 75 directly and ignoring the players who left.
- Counting 12 as the answer instead of adding the three who left.
- Forgetting that each of the three had played three matches.
Exam tip
Separate the matches of the players who left, then use r(r-1)/2 for the rest.
Sources
- NCERT Class XI Mathematics — Permutations and Combinations
- R. S. Aggarwal — Quantitative Aptitude for Competitive Examinations (Permutations and Combinations)