In a certain cricket tournament, 45 matches were played. Each team played…

Concept

When every pair among n teams plays exactly once, each match corresponds to one unordered pair of teams.

Therefore, the total number of matches is n(n − 1)/2.

Application

1. Let n be the number of teams. The given total is 45 matches, so n(n − 1)/2 = 45.

2. Multiply both sides by 2: n(n − 1) = 90.

3. Expand and rearrange: n2 − n − 90 = 0.

4. Factor the quadratic: (n − 10)(n + 9) = 0.

5. Thus n = 10 or n = −9. A number of teams must be positive, so n = 10.

Cross-check

With 10 teams, the number of distinct pairs is 10 × 9/2 = 45, exactly the stated number of matches. Hence, 10 teams participated.