Counting Principles: Permutations

Permutations require students to visualize order-dependent arrangements, a skill that benefits from active, hands-on exploration. Moving beyond abstract formulas, students need to manipulate objects, debate scenarios, and map choices to truly grasp why multiplication and factorial notation work for counting permutations.

• Grade 12
• Mathematics
• 3 activities
• 25 min–45 min

Learning Objectives

1. Calculate the number of possible ordered arrangements of a set of distinct items using the permutation formula.
2. Differentiate between permutations with distinct items and permutations with repeated items, justifying the choice of formula.
3. Apply the fundamental counting principle to solve problems involving ordered selections.
4. Analyze scenarios to determine if order matters and select the appropriate counting strategy.
5. Construct and justify a counting strategy for complex ordered arrangement problems.

Inquiry Circle

⏱ 45 min·Small Groups

Inquiry Circle: The Great Arrangement

Groups are given sets of physical objects (like colored blocks). They must find all possible ways to arrange them under different rules (e.g., 'red must be first' or 'order doesn't matter'). They then derive the formulas based on their findings.

Prepare & details

Explain when the order of selection changes the fundamental counting principle being applied.

Facilitation Tip: During The Great Arrangement, provide physical objects like books or cards so students can physically rearrange them to see the impact of order on the total arrangements.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Think-Pair-Share

⏱ 25 min·Pairs

Think-Pair-Share: Permutation or Combination?

Present 10 real-world scenarios (e.g., lottery numbers, race results, pizza toppings). Students work in pairs to categorize each as a permutation or combination, justifying their choice based on whether order changes the outcome.

Prepare & details

Differentiate between permutations with distinct items and permutations with repeated items.

Facilitation Tip: In Think-Pair-Share, assign each pair a scenario that is clearly a permutation or combination, forcing them to debate the difference using the scenario’s context.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Stations Rotation

⏱ 40 min·Small Groups

Stations Rotation: Pascal's Patterns

Stations explore different aspects of Pascal's Triangle, such as finding combinations, the binomial theorem, and hidden patterns. Students work together to see how the triangle provides a visual shortcut for complex counting problems.

Prepare & details

Construct a counting strategy for scenarios involving ordered arrangements.

Facilitation Tip: At Pascal’s Patterns stations, have students build Pascal’s Triangle from small cases to observe how combinations and permutations relate to its structure.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Teaching This Topic

Experienced teachers approach permutations by first grounding the concept in tangible, relatable contexts like arranging items on a shelf or assigning roles in a club. They avoid jumping straight to formulas, instead using diagrams and small-scale examples to build intuition. Research shows that students grasp the multiplication principle more deeply when they see it emerge from listing outcomes rather than memorizing it as a rule.

What to Expect

Successful learning looks like students confidently distinguishing permutations from combinations, correctly applying the fundamental counting principle, and using factorial notation without confusion. They should articulate why order matters in a given scenario and justify their calculations with clear reasoning.

Watch Out for These Misconceptions

Common Misconception

During Think-Pair-Share: Permutation or Combination?, watch for students labeling a 'combination lock' scenario as a combination in the mathematical sense.

What to Teach Instead

Hand each pair a 'combination lock' scenario card and ask them to label it as a permutation or combination, then physically demonstrate why the order of digits matters by turning the dial.

Common Misconception

During Collaborative Investigation: The Great Arrangement, watch for students defaulting to addition when faced with multiple choice steps.

What to Teach Instead

Ask groups to draw a tree diagram for a small version of their arrangement problem, such as arranging 2 out of 3 books, to visualize why multiplication is used for sequential choices.

Assessment Ideas

Quick Check

After The Great Arrangement, present students with three scenarios: (1) choosing 3 books from 10 to arrange on a shelf, (2) forming a 3-digit number using digits 1, 2, 3 without repetition, and (3) arranging the letters in the word 'APPLE'. Ask students to identify which scenarios involve permutations and why, and to write the formula they would use for each.

Exit Ticket

After Think-Pair-Share: Permutation or Combination?, give students the problem: 'A club has 15 members. How many ways can a president, vice-president, and treasurer be selected?' Ask students to show their calculation using the appropriate permutation formula and to briefly explain why order matters in this situation.

Discussion Prompt

During Station Rotation: Pascal's Patterns, pose the question: 'How does the presence of repeated items, like in the word 'BANANA', change the way we calculate the number of possible arrangements compared to a word with all unique letters, like 'ORANGE'?' Facilitate a discussion where students articulate the need for a different formula and explain its logic.

Extensions & Scaffolding

• Challenge students to create a real-world permutation problem with a twist, such as arranging letters in a word with repeated letters, and solve it using the correct formula.
• For students who struggle, provide a partially filled tree diagram or a scaffolded worksheet that breaks the problem into smaller steps with guided calculations.
• Deeper exploration: Have students research and present how permutations are used in cryptography or sports ranking systems, connecting the math to real applications.

Key Vocabulary

Permutation	An arrangement of objects in a specific order. The order in which items are selected or arranged is important.
Fundamental Counting Principle	If there are 'm' ways to do one thing and 'n' ways to do another, then there are m x n ways to do both. This principle extends to multiple events.
Factorial	The product of all positive integers up to a given integer, denoted by an exclamation mark (e.g., 5! = 5 x 4 x 3 x 2 x 1). Used in permutation calculations.
Permutation with Repetition	A permutation where some items are identical. The formula adjusts to account for the repeated items, preventing overcounting.