<p>A person tries to form as many different parties as he can out of his 20 friends. Each party should consist of the same number. How many friends should be invited at a time? In how many of these parties would the same friends be found?</p>
Step-by-Step Solution
Key Concept: To maximize the number of distinct parties with equal size, we need to find the party size k where ²⁰Cₖ is maximum. For even n, this occurs at k = n/2, giving ²⁰C₁₀ parties. However, the question asks for parties where 'same friends are found'—meaning we count complementary selections (inviting k friends = not inviting 20-k friends) as identical, reducing distinct parties to ²⁰C₁₀/2 = ¹⁹C₉.
<p><strong>Step 1:</strong> For maximum parties with equal size from 20 friends, use the property that ²⁰Cₖ is maximum when k = 10.</p><p><strong>Step 2:</strong> Maximum value = ²⁰C₁₀ (when inviting 10 friends per party)</p><p><strong>Step 3:</strong> The phrase 'same friends be found' implies complementary parties are identical: choosing 10 to invite ≡ choosing 10 to exclude. These represent the same social grouping.</p><p><strong>Step 4:</strong> Count distinct parties = ²⁰C₁₀/2 = [²⁰C₁₀]/2 = ¹⁹C₉ (using the identity ²⁰C₁₀/2 = ¹⁹C₉)</p><p><strong>Answer:</strong> Invite <strong>10 friends</strong> at a time; number of distinct parties = <strong>¹⁹C₉</strong></p>
Correct Answer: ¹⁹C₉