Permutations & Combinations
Arrangements with restrictions
Grade 11
Question:
<p>Ten guests are to be seated in a row of which three are ladies. The ladies insist on sitting together while two of the gentlemen refuse to take consecutive seats. In how many ways can the guests be seated?</p>
Step-by-Step Solution
Key Concept: Treat the three ladies as a single unit (block method), then use complementary counting to handle the two gentlemen who refuse consecutive seats by subtracting arrangements where they sit together.
<p><strong>Step 1: Apply Block Method for Ladies</strong></p><p>Treat 3 ladies as one block. Now we have: 1 block + 7 gentlemen = 8 units to arrange.</p><p>These 8 units can be arranged in 8! ways.</p><p>The 3 ladies within the block can be arranged in 3! ways.</p><p>Total arrangements with ladies together = 8! × 3! = 40320 × 6 = 241920</p><p><strong>Step 2: Apply Complementary Counting for Gentlemen</strong></p><p>Let the two specific gentlemen be G₁ and G₂. We need to subtract cases where G₁ and G₂ sit consecutively.</p><p>If G₁ and G₂ sit together, treat them as one block: 1 block + 1 ladies-block + 5 other gentlemen = 7 units.</p><p>These 7 units arrange in 7! ways.</p><p>G₁ and G₂ can arrange within their block in 2! ways.</p><p>Ladies arrange within their block in 3! ways.</p><p>Arrangements where G₁ and G₂ are consecutive = 7! × 2! × 3! = 5040 × 2 × 6 = 60480</p><p><strong>Step 3: Apply Complementary Principle</strong></p><p>Valid arrangements = Total with ladies together − Arrangements with G₁G₂ consecutive</p><p>= 241920 − 60480 = <strong>181440</strong></p>
Correct Answer: 181440