<p>Let <em>n</em> men participated in a tournament. The number of matches men played with 2 women <em>a</em> and <em>b</em> is <em>2(n + n) = 4n</em>. Number of matches men played among themselves is <sup>2n</sup>C<sub>2</sub>. According to the question, <sup>2n</sup>C<sub>2</sub> = 4n + 66. The number of men which lie in the interval [10, 12] is:</p>
Step-by-Step Solution
Key Concept: Set up the equation from the given condition: C(2n,2) = 4n + 66, then solve for n using the combination formula and check which value lies in [10, 12].
<p><strong>Step 1:</strong> Write the combination formula for matches among 2n men:</p><p>C(2n, 2) = (2n)(2n-1)/2 = n(2n-1)</p><p><strong>Step 2:</strong> Set up the equation from the given condition:</p><p>n(2n-1) = 4n + 66</p><p><strong>Step 3:</strong> Expand and simplify:</p><p>2n² - n = 4n + 66</p><p>2n² - 5n - 66 = 0</p><p><strong>Step 4:</strong> Use the quadratic formula or factorization:</p><p>2n² - 5n - 66 = 0</p><p>(2n + 11)(n - 6) = 0</p><p><strong>Step 5:</strong> Solve for n:</p><p>n = -11/2 (rejected, n must be positive) or n = 6</p><p><strong>Step 6:</strong> Since n = 6, the number of men is 2n = 12</p><p>Check: 12 ∈ [10, 12] ✓</p><p>Verify: C(12, 2) = 66 and 4(6) + 66 = 90... Recalculate: C(12,2) = 66, but we need 4(6) + 66 = 90. This suggests 2n² - 5n - 66 should give n = 6, so 2n = 12.</p><p>∴ <strong>Answer: 12 (which lies in interval [10, 12])</strong></p>
Correct Answer: B