Combinations and Probabilities by Counting

Lesson 2 of 2: When Order Does Not Matter

In this lesson:

• Count unordered selections as combinations
• Compute probabilities for huge sample spaces

Advanced (+) standard — STEM-track scope.

What You Will Be Able to Do

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

1. Decide whether order matters and choose the right tool
2. Compute combinations
3. Compute probabilities by counting favorable and total

Committees of 3 From 8: How Many?

A club forms a 3-person committee from 8 members.

How many different committees are possible?

You counted ordered selections last lesson. Is a committee ordered?

Recap: Ordered Would Be 8P3 = 336

If order mattered, we'd count ordered lists:

But a committee isn't a ranking. Let's check whether 336 overcounts.

Same Committee, Counted Six Times

{Ana, Ben, Cyd} is the same committee as {Cyd, Ben, Ana} — all 6 orders are one.

Each Committee Was Counted 3! = 6 Times

Every unordered committee of 3 corresponds to ordered lists.

So to get the committees, divide the ordered count by 6.

Divide: 8C3 = 336 / 6 = 56

The actual number of committees:

There are 56 distinct 3-person committees from 8 people.

A Combination Is an Unordered Selection

A combination counts the unordered selections of items from .

• The committee: an unordered selection of 3 from 8

nCr = nPr/r! = n!/(r!(n−r)!)

The general combination formula, two equivalent forms:

• Start from the permutation, divide by
• The extra removes the duplicate orderings

The Decision: Does Order Matter?

The one question that picks your tool:

Does order matter?

• Yes → permutation (rankings, positions, passwords)
• No → combination (committees, hands, toppings)

Paired Example: Officers vs Committee

• Officers (order matters):
• Committee (order doesn't):

Permutation or Combination for This Hand?

You deal a 5-card hand from a deck. Does the order you receive the cards matter?

Decide before advancing — then you'll know which tool to count with.

Now Count Favorable and Total

Counting tools let us find probabilities for huge sample spaces.

In a uniform model: — count both.

When you can't list outcomes, count them with permutations or combinations.

Lottery Odds: Counting With a Combination

Choose 6 numbers from 49; order doesn't matter.

One winning set out of nearly 14 million — about 1 in 14 million.

Card Hand: Favorable / 52C5

The chance of a specific kind of 5-card hand:

A dealt hand is unordered — so total is a combination, and favorable must be too.

Count Favorable and Total the Same Way

The consistency rule:

Count favorable and total by the same method.

• Both unordered (combinations), or both ordered (permutations)
• A quick check: favorable total, and

Your Turn: Compute a Probability, Then Interpret

From 8 people, a 3-person team is chosen at random. Ana is one of the 8.

Find , then interpret it.

Count favorable and total the same way, then say what the answer means.

Two Common Errors to Watch For

Overcounting unordered: using a permutation for a committee
Divide out the reorderings — ask the swap test first.

Mixed methods: favorable ordered but total unordered
Count both the same way, or the ratio is meaningless.

Decide Order, Then Count Both Consistently

✓ Ask does order matter? — no → combination, divide by
✓ removes the duplicate orderings
✓ — count both the same way, then interpret

Next: these counting tools feed the binomial distribution and combinatorics in advanced courses.