jmc

Say you're one of the players. How many matches will you play?

Each player plays 7 matches, one against each of the other 7 players. So what's wrong with the following reasoning: "Each of the eight players plays 7 games, so there are $8\times 7=56$ total games played"?

Suppose two of the players are Alice and Bob. Among Alice's 7 matches is a match against Bob. Among Bob's 7 matches is a match against Alice. When we count the total number of matches as $8\times 7$, the match between Alice and Bob is counted twice, once for Alice and once for Bob.

Therefore, since $8\times 7=56$ counts each match twice, we must divide this total by 2 to get the total number of matches. Hence the number of matches in an 8-player round-robin tournament is $\frac{8\times 7}{2}=\boxed{28}$.