Mathematics · Grade 12 · Data Management and Probability · Term 3

Counting Principles: Permutations

Students apply the fundamental counting principle and permutation formulas to count arrangements where order matters.

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

About This Topic

Permutations and combinations are the foundation of counting theory and probability. This topic teaches students how to determine the number of possible outcomes in a variety of scenarios, from arranging books on a shelf to selecting a committee from a larger group. In Ontario's Grade 12 Data Management course, this involves mastering the fundamental counting principle and understanding the role of factorials.

Students learn the crucial distinction between scenarios where order matters (permutations) and where it does not (combinations). They also explore more complex cases, such as arrangements with identical items or circular permutations. This topic is highly engaging when taught through collaborative investigations and games, where students can physically manipulate objects to 'see' the counting patterns before applying the formulas.

Key Questions

1. Explain when the order of selection changes the fundamental counting principle being applied.
2. Differentiate between permutations with distinct items and permutations with repeated items.
3. Construct a counting strategy for scenarios involving ordered arrangements.

Learning Objectives

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

Before You Start

Introduction to Probability

Why: Students need a basic understanding of probability concepts to appreciate the role of counting principles in determining outcomes.

Basic Operations with Integers

Why: Calculations involving factorials and multiplication require proficiency with integer arithmetic.

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.

Watch Out for These Misconceptions

Common Misconception

Students often think 'combination' means any grouping, regardless of the math definition.

What to Teach Instead

In everyday life, we say 'combination lock,' but that's actually a permutation because the order matters! Using a 'Think-Pair-Share' to debate this specific example helps students fix the mathematical definition in their minds.

Common Misconception

Students struggle with when to add versus when to multiply outcomes.

What to Teach Instead

The 'AND' (multiply) vs 'OR' (add) rule is best taught through tree diagrams. Collaborative work where students draw out small-scale versions of problems helps them see why multiplication represents the branching of choices.

Active Learning Ideas

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.

45 min·Small Groups

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.

25 min·Pairs

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.

40 min·Small Groups

Real-World Connections

• In cryptography, the number of possible passwords or encryption keys can be calculated using permutations, especially when considering character order and repetition.
• Athletic event organizers use permutations to determine the number of ways medals can be awarded for first, second, and third place in races or competitions.
• Computer scientists use permutation principles when analyzing the efficiency of sorting algorithms or determining the number of possible states in a system.

Assessment Ideas

Quick Check

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

Give students a 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

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.

Frequently Asked Questions

What is the main difference between a permutation and a combination?

Order. In a permutation, the order of items matters (like a race where 1st, 2nd, and 3rd are different). In a combination, the order does not matter (like picking 3 people to be on the same team). Permutations always result in a larger number of possibilities for the same set of items.

How do I handle items that are identical in a permutation problem?

You divide the total number of permutations by the factorial of the number of identical items. This 'removes' the arrangements that look exactly the same. For example, in the word 'APPLE,' you divide by 2! because there are two 'P's.

How can active learning help students understand counting techniques?

Active learning, such as using manipulatives or playing counting games, makes the abstract formulas concrete. When students physically move objects and see that 'AB' and 'BA' are the same in a combination but different in a permutation, they develop a conceptual 'gut feeling' for which formula to use.

What is the relationship between combinations and Pascal's Triangle?

Each number in Pascal's Triangle is a combination value (nCr). For example, the 3rd row contains the values for 3C0, 3C1, 3C2, and 3C3. This makes the triangle a powerful visual tool for probability and binomial expansions.