Non-Adjacent Permutations

The Gap Method

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📸 Warm-Up: The Photo Line-Up Puzzle

Year 5 Permutations & Combinations

🎯 The Class Photo Dilemma

7 students (A, B, C and D, E, F, G) line up in a single row for a class photo.

⚠️ Teacher's Strict Condition:

No two of A, B, and C may stand adjacent to each other!

STUDENT ROSTER:

Quiet Crew (Unrestricted)

D

E

F

G

Chatterbox Trio (Must Separate)

A

B

C

🚨 The Intuitive Trap: Complementary Counting?

Can we just calculate: Total − (All 3 Together)?

7! − (3! × 5!) = 5040 − 720 = 4320 ? ❌ WRONG!

Why? Subtracting "all 3 together" still leaves cases where exactly two chatterboxes are adjacent (e.g., A & B together, C separated)!

                💡 The Breakthrough: The Gap Method
              
                Instead of fixing bad arrangements, construct valid ones directly!
              
                  🌟 Golden Law of Non-Adjacent Permutations:
                
                  "Arrange the unrestricted students first, then insert the restricted students into the gaps!"

🧩 The 3-Phase Strategy: Step-by-Step Breakdown

Constructive Permutation

Line-Up Stage Visualiser

Key Insight: Each gap can hold at most 1 chatterbox. Since quiet students stand between the gaps, chatterboxes are guaranteed never to touch!

PHASE 1: Arrange Quiet Crew STEP 1

Permute the 4 unrestricted students (D, E, F, G):

4! = 4 × 3 × 2 × 1 = 24 ways

                PHASE 2: Count Available Gaps
                STEP 2
              
                4 students create 2 ends + 3 in-between spaces:
              
                4 + 1 = 5 available gaps (N items → N + 1 gaps)

                PHASE 3: Insert Chatterbox Trio
                STEP 3
              
                Pick 3 distinct gaps from 5 for students A, B, and C:
              
                P(5, 3) = 5 × 4 × 3 = 60 ways

🏆 Final Assembly & Multiplication Principle

Connecting the Independent Steps

⚙️ The Multiplication Principle

Because the line-up is completed through two sequential, independent stages, we multiply the possibilities:

Stage 1: Arrange Quiet Crew (D, E, F, G) 24 ways

Stage 2: Insert Chatterboxes (A, B, C) 60 ways

TOTAL VALID PHOTO LINE-UPS

24 × 60 = ? 1,440 ways

Mental math tip: 24 × 6 = 144 → 144 × 10 = 1,440

                  📋 Master Recipe: Non-Adjacent Problems
                
1

                      Classify: Identify the unrestricted elements vs restricted (non-adjacent) elements.
                    
2

                      Base Line: Arrange the M unrestricted elements first (M!).
                    
3

                      Count Gaps: M elements create exactly (M + 1) available gaps.
                    
4

                      Insert & Multiply: Insert K elements into gaps (P(M+1, K)) and multiply!
                    
                General Formula:
                
                  Total = M! × P(M + 1, K)

              🏅 Mission Accomplished!
            
              You have mastered Non-Adjacent Permutations & The Gap Method!

🚀 After-Class Mastery: Test Your Superpowers!

Independent Challenges • No Answers Revealed

Challenge 1: Photo Line-Up

6 students line up for a photo. Two students, Sam and Leo, must not stand next to each other.

Valid ways:

Challenge 2: Car Parking

5 distinct cars (3 silver, 2 red) park in a row of 5 spaces. The 2 red cars cannot be parked in adjacent spaces.

Valid ways:

Challenge 3: Bookshelf Order

6 distinct books on a shelf: 4 maths textbooks and 2 science textbooks. The 2 science books must never sit side-by-side.

Valid ways:

Challenge 4: Running Lanes (Bonus)

7 sprinters compete in 7 lanes. Runner X and Runner Y must not be assigned to adjacent lanes.

Valid ways: