PERMUTATION VIA CANVA CODING

Permutations Lesson - Slide 1

📚 What Are Permutations?

Definition

A permutation is an arrangement of objects in a specific order. The key word here is order - changing the order creates a different permutation.

💡 Key Insight: In permutations, ABC is different from BAC. Order matters!

Real-Life Examples

Arranging books on a shelf
Seating arrangements at a dinner table
Creating passwords or PIN codes
Race finishing positions (1st, 2nd, 3rd)
Creating team lineups in sports

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🔤 The Fundamental Concept

Let's Start Simple

Imagine you have 3 letters: A, B, and C. How many different ways can you arrange them?

All Possible Arrangements:

ABC | ACB | BAC | BCA | CAB | CBA

Total: 6 arrangements

The Logic Behind It

For the 1st position: we have 3 choices (A, B, or C)
For the 2nd position: we have 2 choices (remaining letters)
For the 3rd position: we have 1 choice (last letter)

Total arrangements = 3 × 2 × 1 = 6

💡 This multiplication principle is the foundation of all permutation problems!

Permutations Lesson - Slide 3

🔢 Factorial Notation (n!)

To make our lives easier, mathematicians created a special notation called factorial.

n! = n × (n-1) × (n-2) × ... × 2 × 1

Examples:

3! = 3 × 2 × 1 = 6
4! = 4 × 3 × 2 × 1 = 24
5! = 5 × 4 × 3 × 2 × 1 = 120
10! = 10 × 9 × 8 × ... × 2 × 1 = 3,628,800

⚠️ Special Case: By definition, 0! = 1 (this makes mathematical formulas work correctly!)

General Formula for Permutations

The number of ways to arrange n different objects is:

P(n) = n!

Permutations Lesson - Slide 4

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✏️ Practice Question 1

Question:

In how many ways can 5 students be arranged in a row for a photograph?

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🎲 Selecting and Arranging (nPr)

Often, we don't use all objects. We select r objects from n objects and arrange them.

The Formula:

nPr = n! / (n - r)!

Where:

n = total number of objects
r = number of objects to arrange
nPr = number of permutations

Example: Racing Medals

In a race with 8 runners, how many ways can we award Gold, Silver, and Bronze medals?

n = 8 (total runners)
r = 3 (positions to fill)
8P3 = 8! / (8-3)! = 8! / 5!
= (8 × 7 × 6 × 5!) / 5!
= 8 × 7 × 6 = 336 ways

💡 Tip: You can cancel out factorials to simplify calculations!

Permutations Lesson - Slide 6

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✏️ Practice Question 2

Question:

A company has 10 employees. In how many ways can they select a President, Vice President, and Secretary?

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⭕ Circular Permutations

When objects are arranged in a circle, some arrangements that look different in a line are actually the same in a circle.

The Formula:

Circular permutations of n objects = (n - 1)!

Why (n-1)! ?

In a circle, we can fix one person's position (say at "12 o'clock") and arrange the others relative to them. This eliminates rotational duplicates.

Example: 5 people around a circular table = (5-1)! = 4! = 24 ways

💡 Real-life applications: Round table seating, necklace designs, circular relay races

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✏️ Practice Question 3

Question:

In how many ways can 6 friends sit around a circular dining table?

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🚫 Permutations with Restrictions

Sometimes we have conditions or restrictions in our arrangements. Let's see how to handle these!

Example 1: Objects Must Stay Together

Problem: How many ways can we arrange the letters A, B, C, D, E if B and C must be together?

Step 1: Treat B and C as one unit: (BC)
Step 2: Now we have 4 objects: A, (BC), D, E
Step 3: Arrange these 4 objects: 4! = 24 ways
Step 4: B and C can switch within their unit: 2! = 2 ways
Total: 24 × 2 = 48 ways

Example 2: Specific Positions

Problem: Arrange 5 people where person A must be at the beginning.

Step 1: Fix A in the first position (1 way)
Step 2: Arrange the remaining 4 people: 4! = 24 ways
Total: 24 ways

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✏️ Practice Question 4

Challenge Question:

How many 4-digit numbers can be formed using digits 1, 2, 3, 4, 5, 6 (without repetition) where the number must be even?

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📝 Quick Reference Summary

1. Basic Permutation: n! = n × (n-1) × (n-2) × ... × 1

2. Selecting r from n: nPr = n! / (n-r)!

3. Circular Permutations: (n-1)!

Key Things to Remember:

✓ Order matters in permutations
✓ Use factorial notation to simplify calculations
✓ Cancel common factorials in nPr formulas
✓ For circular arrangements, use (n-1)!
✓ Break down restriction problems step by step
✓ Identify if objects must stay together or apart

🎓 Final Advice: Practice is key! The more problems you solve, the easier pattern recognition becomes. Always draw diagrams for complex problems!

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🏆 Final Challenge Question

Comprehensive Question:

In how many ways can 7 books be arranged on a shelf if 2 specific books must NOT be next to each other?

🎉 Congratulations on completing the lesson! You now have the tools to solve permutation problems with confidence.