COSMO · STUDY + TEACHING DECK

Counting

September 13, 2025

A question worth sharing

Combinatorics

Cases, complements, and the discipline of counting every object exactly once.

ThinkCompareExplain

Today’s route

A question becomes an explanation.

Try

Enter through a problem. A partial attempt counts.

Connect

Use case, complement, bijection to name what you notice.

Convince

Build an argument a partner can check.

The goal is a method someone else can use.

The opener

Start with a question.

Opening problem

How many handshakes occur if 12 people each shake hands with every other person once? Defend the division by two.

Think · 2 min

Sketch, test a small case, or make a first guess.

Pair · 3 min

Compare methods. Find one thing you agree on and one question.

Be ready to share your route, even if it is unfinished.

Compare before checking

Put two attempts in conversation.

Explain

Tell a partner how you started. Let them finish before responding.

Question

Which step feels certain? Which step still needs evidence?

Build

Combine one useful idea from each attempt into a new route.

Can you explain your partner’s method in your own words?

Board conversation

Notice before naming.

Look for structure

Count choices from one person's point of view.
Identify what gets counted twice.
Describe one handshake without naming an order.

Make a conjecture

We think ___ because ___. A case that might break this is ___.

What changed? What stayed the same?

Useful language

Now the language has a job.

case

a non-overlapping part of the count

complement

total minus the unwanted outcomes

bijection

a one-to-one matching between sets

Use one of these words to explain something you already noticed.

The core idea

Keep the idea. Explain the conditions.

\binom{n}{k}=\frac{n!}{k!(n-k)!},\qquad 0\le k\le n

The idea

Cases, complements, and the discipline of counting every object exactly once.

What to remember

Counting is bookkeeping with a proof attached.

What information does this method keep?

Worked example

One route, one decision at a time

The problem

How many three-letter strings use A, B, C, D with no repeated letter?

Predict each next step

4 choices for the first position.
3 choices remain for the second.
2 choices remain for the third.
Multiply: 4 x 3 x 2 = 24.

Which step needs a reason, not just a calculation?

See the structure

A counting tree makes the multiplication rule visible

Two first choices × three second choices = six complete paths.

Read the representation

Each complete path represents one outcome. Equal branch counts turn repeated choices into multiplication.

Look together

Point to the part that explains the relationship. Change one input and predict what changes.

What can you see here that was harder to see in words?

Two approaches

Compare the reasoning.

Direct counting

Build the allowed objects.
Best when restrictions are local.
Track why each object appears once.

Complement counting

Count all objects, then subtract bad ones.
Best when bad cases are simpler.
Check that bad cases do not overlap.

Which route can you justify? What would you test first?

Reasoning clinic

Find the step that needs repair.

Investigate these concerns

Count all objects, then subtract bad ones.
Best when bad cases are simpler.
Check that bad cases do not overlap.

Repair together

Name the missing condition. Try a small counterexample. Rewrite one step so it is justified.

A useful correction explains why the original step fails.

Work together

Count one set three ways

Your shared task

Groups count five-person committees from eight students using combinations, complements, and a bijection.

Explore

5 min - method one

Compare

5 min - different method

Defend

4 min - compare what each reveals

Everyone should be able to explain the group’s final claim.

Share → connect

What did the room discover?

Bring your evidence

Share a conjecture, a useful example, or a route that changed.

Connections to investigate

A correct count needs completeness and no overlap.
Symmetry can explain equal case sizes.
A formula is useful only after its objects and choices are named.

Which observation explains the mechanism?

Try → explain → check

Your turn to build the route

The problem

How many four-digit numbers have at least one zero?

Predict each next step

Count all four-digit numbers: 9000.
Count those with no zero: 9 x 9 x 9 x 9 = 6561.
Subtract the complement.
9000 - 6561 = 2439.

Which step needs a reason, not just a calculation?

Convince a skeptic

Turn a pattern into a proof.

The claim

A nonempty finite set has equally many even-sized and odd-sized subsets.

Build the argument

Fix one distinguished element.
Toggle whether it belongs to a subset.
This pairs every even subset with one odd subset and reverses perfectly.

What does each step rely on? Where are the conditions used?

Stretch together

Take the idea somewhere new.

Transfer challenge

How many paths move from (0,0) to (5,3) using only right and up steps? Explain the word arrangement behind your count.

Choose a route

Name the objects, the conditions, and a relationship that might help. Start with one test case.

What transfers from the earlier example? What is different?

Peer review

Check the argument together.

Restate

Explain the other person’s route before evaluating it.

Test

Try a small case or boundary case. Does every step still make sense?

Revise

Fix an assumption, add a reason, or find a second method.

Where did your thinking change?

Beyond this problem

Where might this idea help?

A connection

Probability, algorithms, experimental design, and coding all depend on organizing possibilities without omission or repetition.

Discuss the model

What objects or quantities are represented?
Which relationship from today is preserved?
What assumption would you check before trusting it?

Invent one more situation with the same structure.

Exit ticket

Leave with an explanation.

Question 1

What must cases satisfy before their counts can be added?

Question 2

When is a complement easier than a direct count?

Question 3

Why are handshakes divided by two?

Choose one. Write an answer with a reason, then compare with a partner.

Reflect together

Make your thinking visible.

I used to think…

Name a claim or approach that changed.

Now I can explain…

Give the reason, not only the result.

I still wonder…

Write a question worth bringing to the next circle.

Share one sentence. Ask one follow-up question.

The idea to keep

Counting is bookkeeping with a proof attached.

On the worksheet, write the object being counted before writing any formula.

A good question belongs to everyone in the room.