8. In a cricket championship there are 36 matches. What is the number of teams
participating if each team plays exactly one match with every other team?
a. 8
b. 9
c. 10
d. 11
In a cricket championship with 36 matches where each team plays exactly one match against every other team, the number of participating teams is 9. This round-robin format uses the combination formula \( \frac{n(n-1)}{2} = 36 \) to find \( n \), yielding \( n = 9 \) as the solution.
Combination Formula
The total matches equal the ways to choose 2 teams from \( n \) teams, given by \( \binom{n}{2} = \frac{n(n-1)}{2} \).
Set this equal to 36: \( \frac{n(n-1)}{2} = 36 \), so \( n(n-1) = 72 \).
Solving \( n^2 – n – 72 = 0 \) gives \( n = \frac{1 \pm \sqrt{1 + 288}}{2} = \frac{1 \pm 17}{2} \), yielding \( n = 9 \) (discard negative).
Set this equal to 36: \( \frac{n(n-1)}{2} = 36 \), so \( n(n-1) = 72 \).
Solving \( n^2 – n – 72 = 0 \) gives \( n = \frac{1 \pm \sqrt{1 + 288}}{2} = \frac{1 \pm 17}{2} \), yielding \( n = 9 \) (discard negative).
Option Analysis
Verify each option using the formula:
| Option | Teams (n) | Matches \( \frac{n(n-1)}{2} \) | Correct? |
|---|---|---|---|
| a. 8 | 8 | 28 | No |
| b. 9 | 9 | 36 | Yes |
| c. 10 | 10 | 45 | No |
| d. 11 | 11 | 55 | No |
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Why Round-Robin Works
Each match pairs two unique teams without repetition, avoiding double-counting by dividing by 2. For 9 teams, pairings like Team 1 vs. 2 through 8 total 8 for Team 1, but summing all overcounts, so the formula adjusts precisely.
Answer: b. 9
