Permutations & Combinations
Circular Permutations
Grade 11
Question:
<p>The number of ways, in which 5 boys and 3 girls can be seated on a round table if a particular boy \(B_1\) and a particular girl \(G_1\) never sit adjacent to each other, is</p>
<p>\(7!\)</p>
<p>\(5 \times 6!\)</p>
<p>\(6 \times 6!\)</p>
<p>\(5 \times 7!\)</p>
Step-by-Step Solution
Key Concept: Use complementary counting: subtract arrangements where B₁ and G₁ are adjacent from total circular arrangements. For circular arrangements of n objects, fix one position to account for rotational symmetry, giving (n-1)! total arrangements.
<p><strong>Step 1:</strong> Total circular arrangements of 8 people (5 boys + 3 girls)</p><p>Fix one person's position to eliminate rotational counting: (8-1)! = 7! = 5040</p><p><strong>Step 2:</strong> Calculate arrangements where B₁ and G₁ are adjacent</p><p>Treat B₁ and G₁ as a single block. Now we have 7 units to arrange circularly: (7-1)! = 6!</p><p>B₁ and G₁ can be arranged within their block in 2! = 2 ways</p><p>Adjacent arrangements = 6! × 2 = 720 × 2 = 1440</p><p><strong>Step 3:</strong> Apply complementary counting</p><p>Arrangements where B₁ and G₁ are NOT adjacent = 7! - (6! × 2)</p><p>= 5040 - 1440 = 3600</p><p>∴ Answer: B (3600)</p>
Correct Answer: B