figuring out the number of contestants that took part in the primary qualifications

Question

In a table tennis championship each contestant has to play during the primary qualifications against all other contestants, except for himself (as it is simply impossible to play against yourself). The primary qualifications games are already over. It is known that there were 66 of those games. Knowing this number, is it possible to know the number of contestants?

Answer 1

Let's say there were $n$ contestants. Every one of those had to play $n-1$ matches, and every match was between two contestants. Hence there were $\frac{n(n-1)}{2}$ matches played. Setting $\frac{n(n-1)}{2} = 66$ yields $n=12$ or $n=-11$. Since the number of contestants had to be positive, there were $12$ of them.

Answer 2

Hint: if there were $n$ contestants and each one played $n-1$ games, then the total number of games played is $\frac12 n(n-1)$.