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?

Let the total number of participants at the beginning be n. The tournament followed a round robin format where everyone plays everyone else. If no one had left, the total matches would be nC2, which is n multiplied by n minus 1, all divided by 2. In this case, 3 players fell ill. Each of these 3 players played exactly 3 matches. This gives a total of 9 matches involving the ill players. However, we must check if any of these 3 players played against each other. Since each of the 3 ill players played only 3 matches, and they had to play everyone else in the tournament, it is logical to assume their 3 matches were unique or against the remaining healthy players. If they played against each other, those matches are counted in the 9. The remaining n minus 3 players were healthy and played a full round robin among themselves. The number of matches between the healthy players is n minus 3 C2. Total matches = Matches by ill players + Matches among healthy players 75 = 9 + n minus 3 C2 66 = n minus 3 multiplied by n minus 4 divided by 2 132 = n minus 3 multiplied by n minus 4 We need two consecutive integers whose product is 132. These are 12 and 11. n minus 3 = 12 n = 15 Therefore, the total number of participants at the beginning was 15. Correct option is D.