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Teacher Guide: Introduction to Combinations

Learn how to count selections where order doesn't matter using the combinations formula.

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Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Counting Principles. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Define combinations and distinguish them from permutations
  • Apply the combination formula C(n,r)C(n,r) to solve counting problems
  • Simplify factorial expressions efficiently
  • Solve real-world problems involving selections
Prerequisites
  • • Understanding of factorials and factorial notation
  • • Basic knowledge of permutations
  • • Comfort with fraction simplification
Discussion Starters
  • 1. Why do you think lottery odds are so low?
  • 2. If you're choosing pizza toppings, does the order you pick them matter?
  • 3. How is selecting team members different from ranking them?
  • 4. Why does dividing by r!r! give us the number of unique selections?
Common Misconceptions

Thinking a combination count can be bigger than the permutation count for the same nn and rr

Remediation: Show that C(n,r)=P(n,r)÷r!C(n,r) = P(n,r) \div r!: every selection of rr items can be ordered in r!r! ways, so C(n,r)C(n,r) is never larger than P(n,r)P(n,r). They are equal only when r=0r = 0 or r=1r = 1.

Not recognizing when a problem is about combinations

Remediation: Ask: 'If I rearrange my selection, is it still the same outcome?' If yes, it's a combination.

Differentiation Ideas

For Struggling Students:

  • • Start with small numbers like C(4,2)C(4,2) and list all possibilities
  • • Use physical objects (cards, tokens) to visualize selections
  • • Provide a step-by-step formula template

For On-Level Students:

  • • Solve problems with larger numbers requiring calculator use
  • • Compare combination and permutation results for same values
  • • Apply to probability calculations

For Advanced Students:

  • • Explore Pascal's Triangle and its connection to combinations
  • • Solve problems combining permutations and combinations
  • • Investigate the binomial theorem
Standards Alignment
  • HSS.CP.B.9 (CCSS.MATH.CONTENT.HSS.CP.B.9)

    Use permutations and combinations to compute probabilities of compound events

  • HSA.APR.C.5 (CCSS.MATH.CONTENT.HSA.APR.C.5)

    Know and apply the Binomial Theorem, with coefficients found for example from Pascal's Triangle

Lesson Resources
  • visualCombination vs Permutation Explorer

    Compare outcomes when order matters vs doesn't matter

  • activityCommittee Selection Simulator

    Build committees and see why order doesn't matter

  • worksheetReal-World Combinations

    Practice with lottery, cards, and team problems

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

A combination is a selection of items where the order does not matter.
If you're choosing rr items from a group of nn items, the number of combinations is:
C(n,r)=(nr)=n!r!(n−r)!C(n,r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}
Key difference from permutations:
  • Permutation: Order matters (arranging)
  • Combination: Order doesn't matter (selecting)
Example: Choosing 2 people from Alice, Bob, and Carol:
  • {Alice, Bob} is the same as {Bob, Alice}
  • There are only 3 combinations: {A,B}, {A,C}, {B,C}

Worked Examples

A club has 5 members. How many ways can you choose a 2-person committee?
1

Identify n and r

n=5n = 5 (total members), r=2r = 2 (to choose) → n=5n = 5, r=2r = 2
2

Apply the combination formula

C(5,2)=5!2!(5−2)!=5!2!×3!C(5,2) = \frac{5!}{2!(5-2)!} = \frac{5!}{2! \times 3!} → Formula set up
3

Calculate factorials

5×4×3!2×1×3!=5×42\frac{5 \times 4 \times 3!}{2 \times 1 \times 3!} = \frac{5 \times 4}{2} → 202\frac{20}{2}
4

Simplify

202=10\frac{20}{2} = 10 → 1010 ways

Common Mistakes

Confusing combinations with permutations

Why it's wrong: Permutations count arrangements (order matters), combinations count selections (order doesn't matter). P(5,2)=20P(5,2) = 20 but C(5,2)=10C(5,2) = 10.

Correct: Ask yourself: 'Does rearranging change the outcome?' If choosing a committee, {A,B} = {B,A}, so use combinations.

Forgetting to divide by r!r! in the formula

Why it's wrong: The division by r!r! removes the duplicate orderings of the same selection.

Correct: Always use the full formula: C(n,r)=n!r!(n−r)!C(n,r) = \frac{n!}{r!(n-r)!}

Calculating factorials incorrectly

Why it's wrong: Large factorials can be overwhelming. Remember that most terms cancel out.

Correct: Simplify before multiplying: 7!5!×2!=7×62=21\frac{7!}{5! \times 2!} = \frac{7 \times 6}{2} = 21

Why It Matters

Combinations are essential for understanding probability and making real-world decisions:
  • Lottery: How many ways can 6 numbers be drawn from 49?
  • Team Selection: How many ways can you pick 5 players from 12 for a starting lineup?
  • Menu Planning: How many ways can you choose 3 side dishes from 10?
  • Science: How many ways to select samples for experiments?
Understanding combinations helps you calculate probabilities and make informed choices!

Real World Applications

Sports Team Selection

Coaches use combinations to understand how many different team lineups are possible.

Example:

A basketball coach choosing 5 starters from 12 players: C(12,5)=792C(12,5) = 792 possible lineups.

1Try It Yourself

A soccer coach has 18 players and needs to choose 11 for the starting lineup.

How many different starting lineups are possible?

Step 1: Write the mathematical expression

Use C(18,11)C(18, 11):

Card Games

Combinations calculate the number of possible hands in card games.

Example:

A 5-card poker hand from 52 cards: C(52,5)=2,598,960C(52,5) = 2{,}598{,}960 possible hands.

2Try It Yourself

In a card game, each player is dealt 7 cards from a standard 52-card deck.

How many different 7-card hands are possible? Calculate C(52,7)C(52,7).

Step 1: Write the mathematical expression

Set up the formula:

Menu Planning

Restaurants use combinations to calculate meal options and combo deals.

Example:

Choosing 3 appetizers from 10 options: C(10,3)=120C(10,3) = 120 different selections.

3Try It Yourself

A catering company offers 15 different dishes. A customer wants to select 4 for their event.

How many different 4-dish selections are possible?

Step 1: Write the mathematical expression

Calculate C(15,4)C(15, 4):

Key Takeaways

  • 1Combinations count selections where order doesn't matter
  • 2Formula: C(n,r)=n!r!(n−r)!C(n,r) = \frac{n!}{r!(n-r)!}, read as 'n choose r'
  • 3Key difference: Permutations = arrangements (order matters), Combinations = selections (order doesn't)
  • 4C(n,r)=C(n,n−r)C(n,r) = C(n, n-r): Choosing what to include equals choosing what to exclude

Frequently Asked Questions

When do I use combinations vs permutations?

Use combinations when order doesn't matter (selecting a committee, choosing toppings). Use permutations when order matters (assigning positions, arranging a lineup in order).

Why is C(n,r)=C(n,n−r)C(n,r) = C(n, n-r)?

Choosing rr items to include is the same as choosing n−rn-r items to exclude. For example, C(5,2)=C(5,3)=10C(5,2) = C(5,3) = 10.

What does the '!' mean in the formula?

The exclamation mark means factorial. 5!=5×4×3×2×1=1205! = 5 \times 4 \times 3 \times 2 \times 1 = 120. It counts all possible arrangements of a set.

Glossary

Combination
A selection of items where order does not matter
Factorial
The product of all positive integers up to n, written as n!n!
Binomial coefficient
Another name for C(n,r)C(n,r), written as (nr)\binom{n}{r}
n choose r
Common way to read C(n,r)C(n,r), meaning 'the number of ways to choose r from n'

Formula Card

Combinations Formula

C(n,r)=(nr)=n!r!(n−r)!C(n,r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}

n = total items, r = items chosen. Symmetry: $C(n,r) = C(n, n-r)$. Special cases: $C(n,0) = C(n,n) = 1$

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