💺 Seating & Restrictions

Problems involving arranging people or objects with adjacency, spacing, or other constraints.

🎯 Recognition Cues

  • Keywords: “sit together”, “adjacent”, “next to”, “between”, “spacing”
  • Setup: People around a table, in a line, or in specific positions
  • Constraints: Who can sit where, who must be together, who must be apart

📋 Solution Template

  1. Identify the constraint type:

    • Adjacency (must sit together)
    • Spacing (must be apart)
    • Fixed positions
    • Gender/role restrictions

  2. Apply the appropriate method:

    • Adjacency: Treat as a single unit, then arrange
    • Spacing: Use gaps method
    • Fixed positions: Arrange the rest
    • Mixed constraints: Combine methods

  3. Check for overcounting:

    • Circular vs. linear arrangements
    • Indistinguishable objects
    • Reflection symmetry

💡 Micro-Examples

Adjacency Problem

  • Problem: How many ways can you seat 5 people so A and B sit together?
  • Solution: Treat A and B as one unit: $4! \cdot 2! = 48$ ways

Spacing Problem

  • Problem: How many ways can you seat 5 people so A and B are not adjacent?
  • Solution: Total - Adjacent = $5! - 4! \cdot 2! = 120 - 48 = 72$ ways

Gaps Method

  • Problem: How many ways can you seat 3 men and 3 women so no two men sit together?
  • Solution: Arrange women first: $3!$ ways, then place men in 4 gaps: $P(4,3) = 24$ ways. Total: $3! \cdot 24 = 144$ ways

⚠️ Common Pitfalls & Variants

Pitfall: Forgetting to arrange the constrained group

  • Wrong: “A and B sit together” = $4!$ ways (forgot to arrange A and B)
  • Right: $4! \cdot 2!$ ways (arrange the unit, then arrange A and B within)

Pitfall: Circular vs. linear arrangements

  • Wrong: “Around a table” = $n!$ ways
  • Right: $(n-1)!$ ways (fix one person, arrange the rest)

Pitfall: Reflection symmetry

  • Wrong: “On a bracelet” = $(n-1)!$ ways
  • Right: $\frac{(n-1)!}{2}$ ways (divide by 2 for reflection)

🏆 AMC-Style Worked Example

Problem: How many ways can you seat 6 people around a round table so that no two men sit together, given that there are 3 men and 3 women?

Solution:

  • Step 1: Arrange the 3 women around the table: $(3-1)! = 2! = 2$ ways
  • Step 2: This creates 3 gaps between women for the men
  • Step 3: Place the 3 men in these 3 gaps: $3! = 6$ ways
  • Step 4: Total: $2 \cdot 6 = 12$ ways

Key insight: Use the gaps method for spacing constraints in circular arrangements!

🔗 Related Topics

Next: Word Rearrangements → Committees & Conditions