Non-Adjacent Permutations

The Gap Method

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📸 Warm-Up: The Photo Line-Up Puzzle

🎯 The Class Photo Dilemma

7 friends in Year 5 (Emma, Chloe, Zoe and Leo, Sam, Max, Ben) line up in a single row for a class photo.

⚠️ Strict Condition:

Emma, Chloe, and Zoe cannot stand next to each other because they will whisper and giggle!

STUDENT ROSTER:

Quiet Crew (4 Boys)

Leo

Sam

Max

Ben

Chatterboxes (3 Girls)

Emma

Chloe

Zoe

🚨 The Intuitive Trap: Can We Just Subtract?

Could we just find all arrangements and subtract when the 3 girls stand together?

❌ DOES NOT WORK!

                💡 The Gap Method
              
                Instead of fixing messy mistakes, construct the line-up directly:
              
                  🌟 Golden Law of Non-Adjacent Problems:
                
                  "Line up the boys first to create protective dividers, then let the girls choose their gaps!"

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

Line-Up Stage Visualiser

Key Insight: Each gap can hold at most 1 chatterbox. Since boys stand between the gaps, the girls are physically separated!

STEP 1: Line Up the 4 Boys STAGE 1

Assign the 4 boys to the 4 photo spots choice-by-choice:

1st spot: 4 choices • 2nd: 3 • 3rd: 2 • 4th: 1

4 × 3 × 2 × 1 = 24 ways

                STEP 2: Count the Available Gaps
                STAGE 2
              
                The 4 boys create 2 outside ends + 3 spaces between them:
              
                4 boys in a line ⟹ 4 + 1 = 5 available gaps

                STEP 3: Girls Choose Gaps One by One
                STAGE 3
              
                • Emma chooses first: 5 gap choices

                • Chloe chooses second: 4 choices left

                • Zoe chooses last: 3 choices left
              
                5
                × 4
                × 3
                = 60 ways

🏆 Final Assembly & Multiplication Principle

⚙️ The Multiplication Principle

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

Stage 1: Arrange the 4 Boys (4 × 3 × 2 × 1) 24 ways

Stage 2: Girls Choose Gaps (5 × 4 × 3) 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 who can stand anywhere (Boys) vs who cannot touch (Girls).
                    
2

                      Seat the Boys First: Count choices for boys (4 × 3 × 2 × 1 = 24).
                    
3

                      Count Gaps: 4 boys create 4 + 1 = 5 available gaps.
                    
4

                      Girls Choose Gaps: Pick from remaining gaps (5 × 4 × 3 = 60) and multiply!
                    
                Master Rule:
                
                  Total = (Ways for Boys) × (Ways for Girls in Gaps)

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

🚀 After-Class Mastery: Test Your Superpowers!

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: