Lesson 20.3

Permutations and Combinations

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

Introduction

In how many ways can you arrange 3 people in a line? How many ways can you choose 3 people for a committee? The first is a permutation (order matters), the second is a combination(order doesn't matter). This distinction is crucial in counting problems.

Prerequisite Connection

You understand factorials:

Today's Increment

We learn and and when to use each.

Why This Matters

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!