Home›Lesson Plans›Mathematics›Statistics & Probability · Grade 12

Permutations & Combinations

Twelfth graders learn to count arrangements and selections — permutations (where order matters) and combinations (where it doesn’t) — the counting foundations behind probability, from lottery odds to team lineups.

Grade 12Counting55 minutes1 class periodGradual ReleaseExplicit teaching4 StandardsCommon Core
Start the Lesson
Lesson at a Glance

Everything you need before the bell rings

Learning Objectives

Students will be able to…

  • ✓Count with permutations (order matters).
  • ✓Count with combinations (order doesn’t).
  • ✓Choose the right count.
  • ✓Use factorial formulas.
Essential Question

How many ways can you arrange 3 people in a line? How many ways can you CHOOSE 3 from a group? The answers differ — because sometimes order matters and sometimes it doesn’t.

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Lesson Phases
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Vocabulary Terms
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Standards Aligned
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Interactive Task
Model · Interactive Tool

Permutations vs. Combinations

Project this and change n and r. Compare permutations (order matters) and combinations (order doesn’t) for the same choice!

🔢 Compare permutations & combinationsTry it
n (total items): 5
r (choose): 3

The Lesson · Gradual Release (I Do · We Do · You Do)

55 minutes, five moves

Tap any phase to open the teacher moves and student actions.

1

Launch — Arrange or Choose?

5 min

Arranging 3 people in a line vs. choosing 3 from a group — different counts?

👩‍🏫 Teacher Moves

  • Pose an arrange and a choose problem.
  • Ask if they differ.
  • Set the goal: counting.

🎒 Student Actions

  • Consider both.
  • Predict the difference.
  • Predict the idea.
2

I Do — Order Matters or Not

10 min

Teacher models.

👩‍🏫 Teacher Moves

  • Permutation: order matters.
  • Combination: order doesn’t.
  • Use factorial formulas.

🎒 Student Actions

  • Learn permutations.
  • Learn combinations.
  • Use formulas.
3

We Do — Compare Together

13 min

Class uses Permutations vs. Combinations.

👩‍🏫 Teacher Moves

  • Change n and r.
  • Compare the counts.
  • Explain the difference.

🎒 Student Actions

  • Set n and r.
  • Compare them.
  • Explain it.
4

You Do — On Your Own

12 min

Students count.

👩‍🏫 Teacher Moves

  • Solve permutation problems.
  • Solve combination problems.
  • Choose the right count.

🎒 Student Actions

  • Solve perms.
  • Solve combos.
  • Choose right.
5

Close — Close

5 min

One counting problem.

👩‍🏫 Teacher Moves

  • Give a counting problem.
  • Choose and solve it.
  • Hand out the exit ticket.

🎒 Student Actions

  • Choose the count.
  • Solve it.
  • Complete the exit ticket.
Standards Alignment

Built to the standards you report on

Aligned to the Common Core State Standards for Mathematics (High School · Precalculus).

CCSS
S-CP.9

Use permutations and combinations to compute probabilities.

CCSS
S-CP.8

Apply the general Multiplication Rule.

CCSS
S-MD.1

Define and use random variables.

CCSS
S-CP.1

Describe events as subsets of a sample space.

Differentiation

One lesson, every learner

Multilingual Learners

ELL / EMERGING READERS
  • Order matters vs. not cards.
  • Sentence frame: “Order ___, so use ___.”
  • Act out arrangements.

Support & Access

IEP / 504
  • Start by listing arrangements.
  • Compare small cases.
  • Provide the formulas.

Stretch & Extend

GIFTED / EARLY FINISHERS
  • Compute probabilities with counting.
  • Handle repetition.
  • Solve the birthday problem.
Materials

What to gather

  • 📽️Projector / board
  • 🧮Calculators
  • 💻Permutations vs. Combinations
  • 🃏Cards / counters
  • ✏️Pencils
  • 🎫Exit-ticket slips
Vocabulary

Key terms — hover for a quick definition

permutationan arrangement where order matterscombinationa selection where order doesn’t matterfactorialn! = n × (n−1) × … × 1ₙPᵣpermutations of r from nₙCᵣcombinations of r from narrangementan ordering of itemsselectiona chosen subsetsample spaceall possible outcomes
Evaluate

Exit Ticket

Preview the three formative checks. Tap “Sample answer” to see what mastery looks like — hide them before you print for students.

QUESTION 1
How many ways can you arrange 4 books on a shelf?
24 (4! = 4 × 3 × 2 × 1).
QUESTION 2
Does order matter in a permutation or a combination?
A permutation (order matters); in a combination it does not.
QUESTION 3
How many ways can you choose 2 people from 5 (order doesn’t matter)?
10 (₅C₂).

Going deeper? Counting to probability.

Have students use combinations to compute a real probability — like the chance of being dealt a particular hand of cards. A printable counting sheet is in the Math library.

Study · Flashcards

Study the key terms

Tap a card to flip it, then rate whether you knew it. Built from this lesson’s vocabulary.

🃏 Permutations & CombinationsFlip
Card 1
Term
Tap to flip →
Meaning
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Practice · Quiz

Check your understanding

A quick self-check with instant feedback, drawn from this lesson’s key terms.

📝 Permutations & CombinationsQuiz
Score: 0
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Question 1
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Practice · Worksheet

Printable worksheet

A print-and-go review sheet with a built-in answer key. Tap “Show answer key” to reveal answers, or print the clean version for students.

🖨️ Permutations & CombinationsPrint
Name: ________________________
Date: ____________

Part A · Write the word that matches each meaning

Word bank: arrangement, combination, factorial, permutation, sample space, selection, ₙCᵣ, ₙPᵣ
  1. combinations of r from n
  2. an arrangement where order matters
  3. n! = n × (n−1) × … × 1
  4. a chosen subset
  5. permutations of r from n
  6. a selection where order doesn’t matter
  7. an ordering of items
  8. all possible outcomes

Part B · Show what you learned

  1. How many ways can you arrange 4 books on a shelf?
  2. Does order matter in a permutation or a combination?
  3. How many ways can you choose 2 people from 5 (order doesn’t matter)?
Answer key — Part A: 1) ₙCᵣ · 2) permutation · 3) factorial · 4) selection · 5) ₙPᵣ · 6) combination · 7) arrangement · 8) sample space
Part B: 1) 24 (4! = 4 × 3 × 2 × 1). 2) A permutation (order matters); in a combination it does not. 3) 10 (₅C₂).