University of South Carolina High School Math Contest/1993 Exam/Problem 21


Suppose that each pair of eight tennis players either played exactly one game last week or did not play at all. Each player participated in all but 12 games. How many games were played among the eight players?

$\mathrm{(A) \ }10 \qquad \mathrm{(B) \ }12 \qquad \mathrm{(C) \ }14 \qquad \mathrm{(D) \ }16 \qquad \mathrm{(E) \ }18$


Since each player played in all but 12 games, each player played in the same number of games, say $g$. Since each game was played by 2 players, this means there were $\frac12\cdot 8g = 4g$ games played. However, by the given there were also $g + 12$ games played. Thus $4g = g + 12$, $g = 4$ and the answer is $16 \Longrightarrow \mathrm{(D)}$.