Permutations & Combinations
Circular Arrangements with Constraints
Grade 11
Question:
<p>Five boys and 6 girls must sit around a round table. How many arrangements are possible if Ram (a boy) must always be adjacent to Seeta and Geeta (two girls)?</p>
<p>(A) <span class="math">10! - 2 \times 6!</span></p>
<p>(B) <span class="math">6 \times 8!</span></p>
<p>(C) </p>
<p>(D) </p>
Step-by-Step Solution
Key Concept: Treat Ram with his required neighbors as a constrained block. Fix positions using circular symmetry and count internal arrangements.
<p>Treat Ram with Seeta and Geeta as a single unit. This unit plus 9 other people (4 boys + 4 other girls) makes 10 entities to arrange in a circle. Circular arrangements: <span class="math">(10-1)! = 9!</span>. Within the unit, Ram can be in the middle with Seeta and Geeta on either side: 2 arrangements (SRG or GRS). But since we want Ram adjacent to both, treat Seeta-Ram-Geeta as a fixed block with 2 internal orders. Alternatively: Fix Ram, arrange 11 others, with Seeta and Geeta on adjacent seats. With Seeta and Geeta fixed as neighbors of Ram in the circle: The block has 2 arrangements (SRG or GRS). Entities in circle: <span class="math">(10-1)! = 9!</span> but accounting for unit internal order: <span class="math">2 \times 9! = 2 \times 9!/1</span>. Recount: 11 people total. Fix Ram. Seeta and Geeta must sit on his left and right: 2 ways to assign them. Arrange remaining 8 people in remaining 8 seats: <span class="math">8!</span> ways. Total: <span class="math">2 \times 8! \times 3 = 6 \times 8!</span>.</p>
Correct Answer: B