<p>Twenty-eight games were played in a football tournament with each team playing once against each of the others. How many teams were there?</p>
Step-by-Step Solution
Key Concept: When each team plays every other team exactly once, the total number of games equals C(n,2) = n(n-1)/2, where n is the number of teams. Set this equal to 28 and solve the resulting quadratic equation.
<p><strong>Step 1:</strong> When each of n teams plays every other team exactly once, each game involves choosing 2 teams from n teams. Total games = C(n,2) = n(n-1)/2</p><p><strong>Step 2:</strong> Set up the equation: n(n-1)/2 = 28</p><p><strong>Step 3:</strong> Multiply both sides by 2: n(n-1) = 56</p><p><strong>Step 4:</strong> Expand: n² - n = 56</p><p><strong>Step 5:</strong> Rearrange: n² - n - 56 = 0</p><p><strong>Step 6:</strong> Factor: (n-8)(n+7) = 0</p><p><strong>Step 7:</strong> Solve: n = 8 or n = -7. Since n must be positive, n = 8</p><p><strong>Verification:</strong> C(8,2) = 8×7/2 = 28 ✓</p><p>∴ Answer: 8 teams</p>
Correct Answer: 8