1. The problem is to find the total number of rugby matches played if each team plays every other team exactly once. 2. The formula to calculate the number of matches in a round-robin tournament where each team plays every other team once is: $$\text{Number of matches} = \frac{n(n-1)}{2}$$ where $n$ is the number of teams. 3. This formula works because each of the $n$ teams plays $n-1$ matches (against every other team), but since each match involves two teams, we divide by 2 to avoid double counting. 4. If the number of teams is not given, let's denote it as $n$ and the total matches as $M$: $$M = \frac{n(n-1)}{2}$$ 5. For example, if there are 10 teams: $$M = \frac{10 \times 9}{2} = \frac{90}{2} = 45$$ 6. Therefore, the total number of matches played is $\boxed{\frac{n(n-1)}{2}}$ for $n$ teams.