Permutations & Combinations
Counting Games/Arrangements
Grade 11
Question:
<p>There are <em>m</em> men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by 84, then the value of <em>m</em> is __________.</p>
Step-by-Step Solution
Key Concept: Set up equations based on the number of games: men play among themselves (each pair plays 2 games), and men play against women (each man-woman pair plays 2 games). The difference equals 84.
<p><strong>Step 1:</strong> Count games among m men.</p><p>Number of pairs of men = C(m,2) = m(m-1)/2</p><p>Each pair plays 2 games, so games among men = 2 × m(m-1)/2 = m(m-1)</p><p><strong>Step 2:</strong> Count games between men and women.</p><p>There are m men and 2 women. Number of (man, woman) pairs = m × 2 = 2m</p><p>Each pair plays 2 games, so games between men and women = 2 × 2m = 4m</p><p><strong>Step 3:</strong> Set up the difference equation.</p><p>Games among men - Games between men and women = 84</p><p>m(m-1) - 4m = 84</p><p>m² - m - 4m = 84</p><p>m² - 5m = 84</p><p>m² - 5m - 84 = 0</p><p><strong>Step 4:</strong> Solve the quadratic equation.</p><p>(m - 12)(m + 7) = 0</p><p>m = 12 or m = -7</p><p>Since m must be positive, m = 12</p><p><strong>Verification:</strong> Games among men = 12(11) = 132; Games between men and women = 4(12) = 48; Difference = 132 - 48 = 84 ✓</p><p>∴ Answer: <strong>12</strong></p>
Correct Answer: 12