COSMO · STUDY + TEACHING DECK

Counting

November 12, 2025

A question worth sharing

Binomial Coefficients

Combinations count choices, paths, and coefficients through the same recurrence.

ThinkCompareExplain

Today’s route

A question becomes an explanation.

Try

Enter through a problem. A partial attempt counts.

Connect

Use combination, Pascal recurrence, binomial expansion 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 five-person committees can be chosen from 12 students?

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

Try a small case and record every step.
Separate what changes from what stays fixed.
State the pattern in a sentence before naming the theorem.

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.

combination

unordered selection

Pascal recurrence

$\binom{n}{k}$ = $\binom{n-1}{k-1}$ + $\binom{n-1}{k}$

binomial expansion

coefficients of (x+y)^n

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

The core idea

Keep the idea. Explain the conditions.

\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k}

The idea

Combinations count choices, paths, and coefficients through the same recurrence.

What to remember

One choice count can wear several mathematical disguises.

What information does this method keep?

Worked example

One route, one decision at a time

The problem

Explain why $\binom{8}{3}$ = $\binom{8}{5}$ .

Predict each next step

Choosing three included people determines five excluded people.
Choosing five excluded people determines three included people.
The correspondence is reversible.
Therefore the counts are equal.

Which step needs a reason, not just a calculation?

See the structure

The structure behind binomial coefficients

\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k}

For 1 at most k at most n-1, split selections by whether they include a fixed element.

Read the representation

Combinations count choices, paths, and coefficients through the same recurrence.

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.

Structural method

Name the objects and conditions.
Use a reusable relationship.
Explain why the method must work.

Brute force

May help with a small test case.
Can hide the general reason.
Becomes unreliable when the scale grows.

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

Reasoning clinic

Find the step that needs repair.

Investigate these concerns

May help with a small test case.
Can hide the general reason.
Becomes unreliable when the scale grows.

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

Investigate binomial coefficients

Your shared task

Students model one coefficient as a subset count, a lattice path count, and a Pascal triangle entry.

Explore

4 min - test cases

Compare

5 min - form a rule

Defend

4 min - defend the rule

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

One choice count can wear several mathematical disguises.
A complete solution names the condition that makes the method valid.
A counterexample is useful when it exposes a missing condition.

Which observation explains the mechanism?

Try → explain → check

Your turn to build the route

The problem

Find the coefficient of $x^{4}$ in (1 + x)^9.

Predict each next step

The $x^{4}$ term comes from choosing x from four factors.
Choose 4 of the 9 factors.
The coefficient is $\binom{9}{4}$ .
$\binom{9}{4}$ = 126.

Which step needs a reason, not just a calculation?

Convince a skeptic

Turn a pattern into a proof.

The claim

$\binom{n}{k}$ = $\binom{n-1}{k-1}$ + $\binom{n-1}{k}$ .

Build the argument

Fix one distinguished element.
Count selections that include it and those that do not.
The cases are disjoint and complete.

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

Stretch together

Take the idea somewhere new.

Transfer challenge

Prove that the sum of row n of Pascal's triangle is $2^{n}$ .

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

Binomial coefficients organize probability, algebraic expansions, coding, and discrete paths.

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

How many five-person committees can be chosen from 12 students?

Question 2

Explain why this claim is true: $\binom{n}{k}$ = $\binom{n-1}{k-1}$ + $\binom{n-1}{k}$ .

Question 3

Prove that the sum of row n of Pascal's triangle is $2^{n}$ .

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

One choice count can wear several mathematical disguises.

Use the matching practice sets to move from direct examples to proof and transfer.

A good question belongs to everyone in the room.