Counting Principle and Permutations

Lesson 1 of 2: Counting Ordered Selections

In this lesson:

• Multiply choices with the Counting Principle
• Count ordered selections as permutations

Advanced (+) standard — STEM-track scope.

What You Will Be Able to Do

By the end of this lesson, you should be able to:

1. Apply the Fundamental Counting Principle to multi-stage processes
2. Compute factorials and recognize as full arrangements
3. Compute permutations and test if order matters

8 Runners, 3 Medals: How Many Ways?

Eight runners finish a race. Gold, silver, and bronze go to the top three.

In how many different ways can the medals be awarded?

Try to set up the count before the next slide — what choices do you have?

Outfits: 3 Shirts × 4 Pants = 12

• For each of the 3 shirts, there are 4 pants
• outfits

The Fundamental Counting Principle Stated

When a process has stages with independent choices:

Multiply the number of choices at each stage.

• Outfits:
• A 3-letter code:

Adding the Stages Is Wrong — Multiply

A common slip: adding the stage counts.

• "3 shirts + 4 pants = 7"? No — that's not the outfit count
• For each shirt, 4 pants →

Multiply for "and then"; add only for a genuine "or".

Arranging n Distinct Items Gives n!

Arrange distinct items in a row:

• choices for the first spot, for the second, and so on
• This product is n factorial, written

Small Factorials: 5! = 120

Factorials grow fast:

• , , ,
• A handy move: cancel before multiplying out

How Many Ways to Arrange 4 Books?

You have 4 different books to line up on a shelf.

How many distinct orders are possible?

Compute it before advancing — which factorial is this?

The Race Uses the Principle Stage by Stage

Back to the medals — it's the Counting Principle, stage by stage:

• Gold: 8 choices
• Silver: 7 remaining
• Bronze: 6 remaining

Counting the Medal Orders Stage by Stage

Eight, then seven, then six — order matters, so each arrangement is distinct.

A Permutation Is an Ordered Selection

A permutation counts the ordered selections of items from .

• The medals: an ordered selection of 3 from 8
• Notation:

nPr = n!/(n−r)! and Why It Works

The general permutation formula:

• For the race:
• The cancels the positions you don't fill

The Order-Matters Test: Swap Two Items

How to tell you need a permutation:

Swap two of your chosen items. Is it a different outcome?

• Gold–silver vs silver–gold: different → order matters
• If swapping changes the result, it's a permutation

Arrange 4 of 7 Books: 7P4 = 840

A second permutation — choosing and ordering 4 of 7 books:

Four positions, choices descending 7, 6, 5, 4 — order matters on a shelf.

Your Turn: Compute and Justify Order

A club elects a president, vice-president, and treasurer from 9 members.

How many ways? And explain why this is a permutation.

Compute the count, then justify in one sentence that order matters.

Two Common Errors to Watch For

Adding the stages: like instead of
Independent "and then" choices multiply; add only for "or".

Ignoring order: treating an ordered selection as unordered
Run the swap test — if swapping two items changes the outcome, it's a permutation.

Multiply the Choices; Ordered Is a Permutation

✓ Counting Principle: multiply the choices at each stage
✓ Arranging distinct items gives
✓ Ordered selection of from :

Next: when order does NOT matter, and using counts to compute probabilities.