Mathematics · Grade 12

Active learning ideas

Counting Principles: Combinations

Active learning helps students internalize the difference between permutations and combinations by moving beyond abstract formulas. When students physically sort and count groups, they confront the moment where order matters versus where it does not, creating lasting clarity. This hands-on engagement reduces reliance on rote memorization and builds intuitive understanding of why division by r! adjusts for overcounting in combinations.

• Ontario Curriculum Expectations
• HSS.CP.B.9

• 30–45 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Collaborative Problem-Solving
• 35 min · Pairs

Sorting Cards: Permutations vs Combinations

Prepare 20 scenario cards, such as 'choose 3 toppings for pizza' or 'arrange 3 books on shelf'. Pairs sort cards into combinations or permutations piles and write justifications. Regroup to share and refine categorizations as a class.

Compare permutations and combinations, identifying when each counting method is appropriate.

Facilitation Tip

During the Sorting Cards activity, circulate and listen for pairs debating whether two lists represent the same group; this is the moment to reinforce the definition of order irrelevance.

What to look for

Present students with two scenarios: Scenario A: Selecting a president, vice-president, and treasurer from a club of 10 members. Scenario B: Selecting a committee of 3 members from a club of 10 members. Ask students to identify which scenario requires combinations and to write the formula they would use to solve it, explaining their choice.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 02

• Collaborative Problem-Solving
• 45 min · Small Groups

Committee Selection Simulation

Give small groups 10 student name cards. Task them to count ways to select committees of 4 or 5 members, first assuming all distinct, then adjusting for identical roles. Groups record formulas and verify with calculators.

Analyze how to adjust counting methods when items in a set are indistinguishable.

Facilitation Tip

In the Committee Selection Simulation, limit the group size to 8 to ensure manageable counts and visible repetition when order is ignored.

What to look for

Pose the question: 'Imagine you have 5 identical red balls and 3 identical blue balls. How would you determine the number of unique ways to arrange these balls in a line?' Facilitate a discussion where students explore strategies for handling indistinguishable items and compare their approaches to the formal combination formula with adjustments.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 03

• Collaborative Problem-Solving
• 40 min · Small Groups

Pascal's Triangle Build

Start with whole class modeling the first rows on board. In small groups, students use grid paper and colored markers to extend to row 10, verifying combinations formula for select entries. Discuss patterns observed.

Justify why Pascal's Triangle is a visual representation of the combinations formula.

Facilitation Tip

When building Pascal's Triangle, have students label each entry with the corresponding C(n,r) value immediately to link visual patterns to combinatorial meaning.

What to look for

Provide students with a partially completed Pascal's Triangle (e.g., up to row 4). Ask them to calculate and fill in the next two rows. Then, ask them to identify which row corresponds to combinations of 6 items taken 0, 1, 2, 3, 4, 5, or 6 at a time, and write the corresponding C(n,r) notation.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 04

• Collaborative Problem-Solving
• 30 min · Individual

Indistinguishable Objects Puzzle

Individuals solve puzzles like counting distinct hands from identical/deck cards. Pairs compare solutions, adjust formulas, and present adjustments. Class votes on correct counts.

Compare permutations and combinations, identifying when each counting method is appropriate.

What to look for

Present students with two scenarios: Scenario A: Selecting a president, vice-president, and treasurer from a club of 10 members. Scenario B: Selecting a committee of 3 members from a club of 10 members. Ask students to identify which scenario requires combinations and to write the formula they would use to solve it, explaining their choice.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

A few notes on teaching this unit

Teach combinations by anchoring them in real scenarios students find relevant, such as team selections or committees. Avoid starting with the formula; instead, let students grapple with counting identical outcomes to appreciate why division by r! is necessary. Research shows that students grasp the subtraction in n-r more easily when they first see why some items are excluded from the group entirely. Emphasize peer explanation over teacher correction to build durable understanding.

By the end of these activities, students will confidently distinguish when to use combinations versus permutations and correctly apply the formula C(n, r) = n! / (r!(n-r)!). They will justify their choices with concrete examples and adjust calculations for indistinguishable items without prompting. Peer discussion and shared corrections will strengthen both accuracy and conceptual reasoning.

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Watch Out for These Misconceptions

• During the Sorting Cards activity, watch for students who treat two ordered lists as different even when they contain the same names, indicating they still confuse order with combination principles.

  Ask them to physically rearrange the cards and recount, then compare totals when order is ignored versus when it matters. Prompt them to explain why identical groups should count once in combinations.

• During the Indistinguishable Objects Puzzle, watch for students who count all arrangements as unique despite identical items, inflating the total.

  Have them swap identical objects within a group and ask if the arrangement has truly changed. Guide them to adjust the formula by dividing by the factorial of the counts of each indistinguishable type.

• During Pascal's Triangle Build, watch for students who see the triangle as a decorative pattern unrelated to combinations.

  After building each row, ask them to calculate C(n,0), C(n,1), C(n,2) for that row and compare results. Connect each entry to a concrete selection task to anchor meaning.

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Methods used in this brief

• Collaborative Problem-Solving