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 beginni

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Q: 67 (IAS/2006)
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?

question_subject: 

Logic/Reasoning

question_exam: 

IAS

stats: 

0,5,3,1,0,2,5

keywords: 

{'tournament': [0, 0, 1, 0], 'other participants': [0, 0, 1, 0], 'matches': [0, 0, 4, 1], 'participants': [0, 0, 2, 1], 'total number': [0, 0, 3, 0], 'match': [1, 0, 0, 0], 'players': [0, 1, 4, 2]}

The question asks for the total number of participants in a tournament where everyone plays each other once. If 3 players fell ill after playing three matches each, that`s a total of 9 matches. These 9 matches must be subtracted from the total of 75, leaving 66 matches played by the remaining players.

Bearing in mind the formula for combinations in a group (n(n-1)/2), where n is the number of participants, is used to calculate the total number of matches in such a tournament.

Option 1: If there were 8 participants, they would have played 28 matches (8(8-1)/2), which is not enough.

Option 2: If there were 10 participants, they would have played 45 matches (10(10-1)/2), still not enough.

Option 3: For 12 participants, the number of matches would be 66 (12(12-1)/2), close but not including the 9 matches played by the ill participants.

Option 4: If there were 15 participants, they would have played 105 matches (15(15-1)/2), but after deducting the matches played by the ill participants (15-3) and recalcul

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