Lesson 20.3

Permutations and Combinations

Distinguishing between arrangements where order matters and selections where it does not.

Introduction

Distinguishing between arrangements where order matters and selections where it does not.

Past Knowledge

You understand factorials:

Today's Goal

We learn and and when to use each.

Future Success

Counting is foundational to probability, cryptography, and computer science algorithms.

Key Concepts

Permutation (Order Matters)

Arranging r items from n distinct items in a specific order.

Combination (Order Doesn't Matter)

Selecting r items from n distinct items (groups, committees, etc.).

The Key Question

Does changing the order create a different outcome?

Yes → Permutation. No → Combination.

Relationship

Combinations are permutations divided by the number of ways to arrange r items.

Worked Examples

Example 1: Race Positions (Basic - Permutation)

10 runners in a race. How many ways for 1st, 2nd, 3rd place?

Order matters (1st ≠ 2nd), so use permutation.

ways

Example 2: Committee Selection (Basic - Combination)

Choose 4 people from 10 for a committee. How many ways?

Order doesn't matter (it's just a group), so use combination.

ways

Example 3: Mixed Problem (Advanced)

A committee of 3 is chosen from 5 men and 4 women. How many committees have exactly 2 women?

Choose 2 women from 4 AND 1 man from 5:

30 committees

Common Pitfalls

Using permutation when order doesn't matter

Choosing a team ≠ arranging a lineup. Teams use combinations.

Forgetting 0! = 1

Adding instead of multiplying

For "AND" situations, multiply. For "OR" situations, add.

Real-World Application

Lottery Odds

In a 6/49 lottery, you choose 6 numbers from 49. Order doesn't matter, so:

That's roughly 1 in 14 million odds!