Introduction
Counting Principle and Permutations is an important Grade 8 math skill because students are moving from simple answers toward explaining how the math works.
In this lesson, students use models, real questions, worked examples, practice problems, and two online quizzes to build confidence with counting principle and permutations.
What Is Counting Principle and Permutations?
Counting Principle and Permutations means choosing a model, naming what each number means, and explaining the strategy.
The goal is not only to get the answer. Students should be able to show the idea, explain the strategy, and check whether the answer makes sense.
Understanding Counting Principle and Permutations
Before solving, students should slow down and decide what each number, shape, unit, or label represents.
- Read the question carefully and identify what is being asked.
- Choose a model, equation, table, or diagram that matches the situation.
- Solve one step at a time and keep units or labels attached.
- Use the answer explanation to check that the result makes sense.
Visual Models
Visual Model 1
Question: A student can wear a shirt, pants, and shoes. There are \(3\) shirt colors, \(2\) pant types, and \(4\) shoe styles. How many different outfits are possible?
| Shirts | Pants | Shoes |
|---|---|---|
| \(3\) colors | \(2\) types | \(4\) styles |
- A. \(3 + 2 + 4 = 9\)
- B. \(3 \times 2 \times 4 = 24\)
- C. \(3^2 \times 4 = 36\)
- D. \(24 + 3 = 27\)
Why it works: Multiply the number of choices at each step: \(3 \times 2 \times 4 = 24\) different outfits.
Answer: \(24\)
Visual Model 2
Question: A probability experiment involves flipping a coin (heads or tails) and rolling a die (1 through 6). How many different outcomes are possible?
- A. \(2 + 6 = 8\)
- B. \(2 \times 6 = 12\)
- C. \(6^2 = 36\)
- D. \(2 + 6 \times 2 = 14\)
Why it works: Coin flip (2 outcomes) combined with die roll (6 outcomes): \(2 \times 6 = 12\) total outcomes.
Answer: \(12\)
Worked Examples
Example 1
Question: A game requires picking a card from a deck of 10 cards and then spinning a spinner with 4 sections. How many possible results are there?
- A. \(10 + 4 = 14\)
- B. \(10 \times 4 = 40\)
- C. \(10 - 4 = 6\)
- D. \(4^{10}\)
- Card pick (10 choices) times spinner section (4 choices): \(10 \times 4 = 40\) possible results.
Answer: \(40\)
Example 2
Question: A tree diagram shows that a choice has 3 branches first, then each branch splits into 2 more branches. How many final outcomes are there?
- A. \(3 + 2 = 5\)
- B. \(3 \times 2 = 6\)
- C. \(3^2 = 9\)
- D. \(2^3 = 8\)
- Each of the 3 initial branches splits into 2 outcomes: \(3 \times 2 = 6\) final outcomes.
Answer: \(6\)
Example 3
Question: In a game, you spin a spinner with 3 colors (red, yellow, green) and flip a coin (heads, tails). The table shows all outcomes. How many outcomes are there?
| Red | Yellow | Green | |
|---|---|---|---|
| Heads | (H,R) | (H,Y) | (H,G) |
| Tails | (T,R) | (T,Y) | (T,G) |
- A. \(3 + 2 = 5\)
- B. \(3 \times 2 = 6\)
- C. \(2^3 = 8\)
- D. \(3^2 = 9\)
- From the table, count all cells: \(3\) colors times \(2\) coin outcomes \(= 6\) total outcomes.
Answer: \(6\)
Real-World Word Problems
Problem 1
Question: How many ways can 4 books be arranged on a shelf?
- A. \(4 + 3 + 2 + 1 = 10\)
- B. \(4^4 = 256\)
- C. \(4! = 24\)
- D. \(\binom{4}{2} = 6\)
Why it works: Arrange all 4 books in order: \(4! = 4 \times 3 \times 2 \times 1 = 24\).
Answer: \(24\)
Problem 2
Question: A teacher selects 2 students from a class of 6 to be line leaders. The first student is the main leader and the second is the assistant. In how many ways can this be done?
- A. \(6 + 5 = 11\)
- B. \(6 \times 5 = 30\)
- C. \(\binom{6}{2} = 15\)
- D. \(6^2 = 36\)
Why it works: Choose a main leader (6 choices) then an assistant from the remaining 5 students: \(6 \times 5 = 30\).
Answer: \(30\)
Common Mistakes
- Rushing before identifying what the numbers represent.
- Choosing an operation that does not match the situation.
- Dropping labels, units, or context from the answer.
- Skipping the estimate or reasonableness check.
Strategy Tips
- Underline the question being asked.
- Use a model before jumping to computation.
- Write an equation that matches the story or picture.
- Explain the final answer in a sentence.
Practice Questions
Question 1
A menu offers \(4\) appetizers, \(6\) entrees, and \(3\) desserts. How many different meal combinations of one appetizer, one entree, and one dessert are possible?
- A. \(13\)
- B. \(24\)
- C. \(72\)
- D. \(120\)
Question 2
A license plate has \(2\) letters followed by \(3\) digits. Letters can be any of A through Z, and digits can be \(0\) through \(9\). How many possible license plates are there?
- A. \(26^2 \times 10^3 = 676{,}000\)
- B. \(26 + 26 + 10 + 10 + 10 = 82\)
- C. \(26 \times 10 = 260\)
- D. \((26 + 10)^5 = 60{,}466{,}176\)
Question 3
A password requires choosing a color and a number from \(1\) to \(9\). There are \(5\) colors available. How many different passwords are possible?
- A. \(5 + 9 = 14\)
- B. \(5 \times 9 = 45\)
- C. \(9^5 = 59{,}049\)
- D. \(\frac{9}{5} = 1.8\)
Question 4
A sandwich shop offers \(2\) types of bread and \(7\) topping options. How many different one-topping sandwiches can be made?
- A. \(2 + 7 = 9\)
- B. \(2 \times 7 = 14\)
- C. \(7^2 = 49\)
- D. \(2 \times 7 \times 7 = 98\)
Question 5
A schedule allows you to pick a morning activity and an afternoon activity from the list below. Morning: hiking or swimming. Afternoon: painting, music, or reading. How many different schedules are possible?
- A. \(2 + 3 = 5\)
- B. \(2 \times 3 = 6\)
- C. \(3^2 = 9\)
- D. \((2+3)^2 = 25\)
Question 6
A security code consists of 3 digits, each from 0 to 9, with no repetition allowed. How many different codes are possible?
- A. \(10^3 = 1{,}000\)
- B. \(10 \times 9 \times 8 = 720\)
- C. \(10!\)
- D. \(\binom{10}{3} = 120\)
Full Answer Explanations Click to show all answers and explanations
Question 1
Answer: \(72\)
By the Counting Principle, multiply choices: \(4\times6\times3=72\).
Question 2
Answer: \(676{,}000\)
Each letter has 26 choices and each digit has 10 choices. Multiply: \(26 \times 26 \times 10 \times 10 \times 10 = 676{,}000\).
Question 3
Answer: \(45\)
For each of the \(5\) color choices, you can pair it with any of the \(9\) numbers. By the Counting Principle, multiply: \(5 \times 9 = 45\) different passwords.
Question 4
Answer: \(14\)
Choose one bread and one topping: \(2 \times 7 = 14\) different sandwiches.
Question 5
Answer: \(6\)
Two morning choices times three afternoon choices: \(2 \times 3 = 6\) possible schedules.
Question 6
Answer: \(720\)
First digit: 10 choices. Second digit: 9 remaining choices. Third: 8 remaining. Total: \(10 \times 9 \times 8 = 720\).
Connection to Standards
This lesson supports Grade 8 math expectations for reasoning, modeling, problem solving, and explaining answers clearly. It connects classroom skills to the kind of questions students see on state math assessments.
Summary
Counting Principle and Permutations becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.
GOLDEN RULE
Understand the model before choosing the operation.
