Permutations and Combinations

🎰 The Magic of Counting: Permutations & Combinations

Imagine you’re at an ice cream shop with 3 flavors. How many different ways can you enjoy them? Let’s find out!

🌟 The Counting Principle: Your First Superpower

What’s the Big Idea?

Think of it like getting dressed. You have 3 shirts and 2 pants. How many outfits can you make?

Shirt 1 → Pants 1 ✓
Shirt 1 → Pants 2 ✓
Shirt 2 → Pants 1 ✓
Shirt 2 → Pants 2 ✓
Shirt 3 → Pants 1 ✓
Shirt 3 → Pants 2 ✓

Answer: 3 × 2 = 6 outfits!

🔑 The Golden Rule

If you can do Task A in m ways and Task B in n ways, you can do BOTH in m × n ways.

Real-Life Example

Building a sandwich:

• 4 types of bread
• 3 types of filling
• 2 types of sauce

Total sandwiches: 4 × 3 × 2 = 24 different sandwiches!

graph TD
    A["Start"] --> B["Pick Bread: 4 ways"]
    B --> C["Pick Filling: 3 ways"]
    C --> D["Pick Sauce: 2 ways"]
    D --> E["4 × 3 × 2 = 24 ways!"]

🔢 Factorial: The Power Multiplier

What’s a Factorial?

A factorial is like a countdown multiplication party! We write it with an exclamation mark: n!

5! means: 5 × 4 × 3 × 2 × 1 = 120

It’s like asking: “In how many ways can 5 friends stand in a line?”

Quick Factorial Table

🎯 Special Case: 0! = 1

Why? Think of it this way: How many ways can you arrange nothing? There’s exactly one way to have nothing—just leave it empty!

Simple Example

How many ways can you arrange the letters A, B, C?

ABC, ACB, BAC, BCA, CAB, CBA = 6 ways

Using factorial: 3! = 3 × 2 × 1 = 6 ✓

🎭 Permutations: When ORDER Matters!

The Core Idea

A permutation is an arrangement where the order is important.

Think about it:

• 🔐 Lock code 1-2-3 is DIFFERENT from 3-2-1
• 🏃 1st, 2nd, 3rd place in a race matter!
• 📞 Phone number 123 ≠ 321

The Formula

Arranging r items from n items:

P(n,r) = n! ÷ (n-r)!

Or think of it as: n × (n-1) × (n-2) × … (r times)

🎬 Movie Example

3 friends competing for 1st and 2nd place (from 5 friends)

P(5,2) = 5 × 4 = 20 ways

Why?

• 5 choices for 1st place
• 4 remaining for 2nd place
• 5 × 4 = 20 different results!

graph TD
    A["5 Friends"] --> B["1st Place: 5 choices"]
    B --> C["2nd Place: 4 choices left"]
    C --> D["Total: 5 × 4 = 20 ways"]

Quick Examples

🤝 Combinations: When ORDER Doesn’t Matter!

The Core Idea

A combination is a selection where order doesn’t matter.

Think about it:

• 🍕 Pizza toppings: Cheese+Pepperoni = Pepperoni+Cheese (same pizza!)
• 👥 Team selection: Picking Amy & Bob = Picking Bob & Amy
• 🎁 Choosing gifts: It’s the same gift set regardless of picking order

The Formula

Choosing r items from n items:

C(n,r) = n! ÷ (r! × (n-r)!)

Or think: P(n,r) ÷ r! (divide permutation by arrangements of selected items)

🍦 Ice Cream Example

Choose 2 scoops from 4 flavors

Using formula:

C(4,2) = 4! ÷ (2! × 2!)
       = 24 ÷ (2 × 2)
       = 24 ÷ 4
       = 6 combinations

The 6 combinations:

1. Vanilla + Chocolate
2. Vanilla + Strawberry
3. Vanilla + Mango
4. Chocolate + Strawberry
5. Chocolate + Mango
6. Strawberry + Mango

🎯 Key Insight: Permutation vs Combination

🏗️ Arrangement Problems: Putting Things in Order

Type 1: Simple Line Arrangements

How many ways can 4 kids stand in a line?

Answer: 4! = 4 × 3 × 2 × 1 = 24 ways

Type 2: Arrangements with Restrictions

5 people in a line, but Mom must be at the end

Solution:

• Fix Mom at one end (1 way)
• Arrange other 4 people: 4! = 24 ways
• Total: 24 ways

Type 3: Circular Arrangements

6 friends sitting around a round table

For circles, we fix one person and arrange the rest: (n-1)! = 5! = 120 ways

Why? Because rotating everyone keeps the same arrangement!

graph TD
    A["Circular Arrangement"] --> B["Fix 1 person"]
    B --> C["Arrange remaining n-1"]
    C --> D["Answer: n-1 factorial"]

Type 4: Arrangements with Identical Items

Arrange the letters in BOOK

• Total letters: 4
• Letter O repeats: 2 times

Formula: 4! ÷ 2! = 24 ÷ 2 = 12 arrangements

Arrange the letters in MISSISSIPPI

• Total: 11 letters
• I repeats: 4 times
• S repeats: 4 times
• P repeats: 2 times

Formula: 11! ÷ (4! × 4! × 2!) = 34,650 arrangements

🎯 Selection Problems: Choosing Without Arranging

Type 1: Simple Selection

Choose a team of 3 from 8 players

C(8,3) = 8! ÷ (3! × 5!)
       = (8 × 7 × 6) ÷ (3 × 2 × 1)
       = 336 ÷ 6
       = 56 teams

Type 2: Selection with Categories

From 5 boys and 4 girls, select 2 boys AND 2 girls

• Ways to pick 2 boys from 5: C(5,2) = 10
• Ways to pick 2 girls from 4: C(4,2) = 6

Total: 10 × 6 = 60 ways

Type 3: At Least / At Most Problems

Select 3 people from 6, with at least 1 woman (3 women, 3 men available)

Strategy: Total ways - All men

• Total: C(6,3) = 20
• All men (no women): C(3,3) = 1

Answer: 20 - 1 = 19 ways

Type 4: Handshakes & Connections

8 people at a party. Everyone shakes hands once. How many handshakes?

Each handshake = selecting 2 people from 8

C(8,2) = (8 × 7) ÷ 2 = 28 handshakes

🧠 Quick Memory Tricks

🔐 When to Use What?

Ask yourself: “Does switching the order create something NEW?”

📝 Formula Cheat

Permutation: P(n,r) = n!/(n-r)!
             "Pick and Arrange"

Combination: C(n,r) = n!/[r!(n-r)!]
             "Just Pick"

🎮 The Restaurant Test

If it’s like ordering courses (appetizer THEN main THEN dessert) → Permutation

If it’s like buffet (grab whatever, order doesn’t matter) → Combination

🎉 Confidence Checkpoint

You now understand:

✅ Counting Principle - Multiply choices together ✅ Factorial - n! = countdown multiplication ✅ Permutations - Order matters, use when arranging ✅ Combinations - Order doesn’t matter, use when selecting ✅ Arrangement tricks - Circles, repeats, restrictions ✅ Selection strategies - Categories, at least/most problems

Next time you see “how many ways,” you’ll know exactly what to do! 🚀