<p>Two women and some men participated in a chess tournament in which every participant played two games with each of the other participants. If the number of games that the men played between themselves exceeds the number of games that the men played with the women by 66, then the number of men who participated in the tournament lies in the interval:</p>
Step-by-Step Solution
Key Concept: Set up equations based on games played: if there are m men, games between men = m(m-1), games between men and women = 2m(2). The constraint gives m(m-1) - 4m = 66, which simplifies to a quadratic equation.
<p><strong>Step 1:</strong> Let the number of men be m.</p><p><strong>Step 2:</strong> Games played between men (each pair plays 2 games): Each pair of men plays 2 games. Total pairs of men = C(m,2) = m(m-1)/2. Total games = m(m-1).</p><p><strong>Step 3:</strong> Games played between men and women (each pair plays 2 games): Each man plays with 2 women, and they play 2 games each. Total games = m × 2 × 2 = 4m.</p><p><strong>Step 4:</strong> Set up the equation: m(m-1) - 4m = 66</p><p><strong>Step 5:</strong> Simplify: m² - m - 4m = 66 → m² - 5m - 66 = 0</p><p><strong>Step 6:</strong> Factor: (m - 11)(m + 6) = 0</p><p><strong>Step 7:</strong> Since m must be positive: m = 11</p><p>∴ The number of men is 11, which lies in the interval containing 11 (typically options like [10, 15] or [8, 12]). Answer: B</p>
Correct Answer: B