<p>A person has \(n\) friends. The minimum value of \(n\) so that a person can invite a different pair of friends every day for four weeks in a row is ___.</p>
Step-by-Step Solution
Key Concept: Four weeks = 28 days, so we need 28 different pairs. The number of ways to choose 2 friends from n friends is C(n,2) = n(n-1)/2, which must be ≥ 28.
<p><strong>Step 1:</strong> Determine the number of days.</p><p>Four weeks = 4 × 7 = 28 days</p><p><strong>Step 2:</strong> Each day requires a different pair of friends.</p><p>Number of distinct pairs from n friends = C(n,2) = n(n-1)/2</p><p><strong>Step 3:</strong> Set up the inequality.</p><p>We need: C(n,2) ≥ 28</p><p>n(n-1)/2 ≥ 28</p><p>n(n-1) ≥ 56</p><p><strong>Step 4:</strong> Solve for minimum n.</p><p>Test values:</p><p>• n = 7: 7(6) = 42 < 56 ✗</p><p>• n = 8: 8(7) = 56 ≥ 56 ✓</p><p><strong>Step 5:</strong> Verify.</p><p>C(8,2) = 8 × 7 / 2 = 28 pairs exactly</p><p>∴ Answer: 8</p>
Correct Answer: 8