23. A chess tournament is planned for teams, each of which has exactly three players. Each player in a team will play exactly once against each player from all the other teams. For organisational reasons, no more than 250 games can be played in total. At most, how many teams can enter the tournament?
A 11
B 10
C 9
D 8
E 7
23. E Let the largest number of teams that can enter the tournament be $n$. Hence there would be $3 n$ players in total. Each player in a team will play a game against every player from all other teams and hence the total number of games played is $\frac{3 n \times(3 n-3)}{2}=\frac{9 n(n-1)}{2}$. Since no more than 250 games can be played, we have $\frac{9 n(n-1)}{2} \leq 250$. Therefore $n(n-1) \leq \frac{500}{9}=55 \frac{5}{9}$. Since $8 \times 7=56>55 \frac{5}{9}$, we have $n<8$. Also, since $7 \times 6=42<55 \frac{5}{9}, n=7$ is a possible solution. Therefore the largest number of teams that can enter the tournament is 7 .