DoAssignment.ca

DoAssignment study guide

P4 · Apply the fundamental counting principle

Learn to apply the fundamental counting principle through clear examples and targeted practice.

Alberta Grade 12 Mathematics

Probability

Count multi-step choices by multiplying the options at each stage

Many choices happen in stages. You might choose a shirt, then pants, or choose one item for each of several positions. Listing every possible result can take a long time. The fundamental counting principle provides a direct way to count complete outcomes. First, name each stage. Then count the options available at that stage. Before multiplying, check whether an earlier choice changes the options that remain.

What you will learn

  • Identify the stages in a multi-step choice.
  • Apply the fundamental counting principle to count complete outcomes.
  • Adjust a stage count when an earlier choice changes what remains available.
  • Explain what a product counts and check that it answers the question.

1. Review: stages and complete outcomes

A stage is one decision in a larger task, such as choosing a drink. An outcome is one result of a choice. A complete outcome includes a choice from every required stage.
Suppose a student chooses one of two notebooks and one of three pens. A complete outcome is one notebook paired with one pen. For each notebook, all three pens are available. So there are three pairs for each notebook, or six complete outcomes.
The question tells you what counts as one outcome. If the question asks for outfits, for example, counting only the shirts does not give the number of outfits.
  • A stage is one part of a multi-step choice.
  • A complete outcome includes a choice from every required stage.

2. The counting rule

The fundamental counting principle applies when a task has two or more stages and you know how many choices are available at each stage. If every choice at one stage can be paired with every choice at the other stages, multiply the stage counts.
For example, if there are aa choices at the first stage and bb choices at the second, the number of complete outcomes is a×ba \times b. For more stages, use one factor for each stage. Each factor counts the available choices at one stage; the product counts complete outcomes.
A slot display is a way to organize the count. Each slot represents a stage, and its number records the choices for that stage. The display can help you notice a missing stage before you calculate.
The number of choices can change from stage to stage. If an earlier choice uses an item that cannot be used again, count only the items still available at the later stage. The rule still uses multiplication; the stage counts must reflect the restrictions.
The principle counts possibilities. It does not say that all outcomes are equally likely, and it does not assign probabilities. Keep the question in view: count complete outcomes that follow the stated rules.
N=n1×n2×⋯×nkN=n_1 \times n_2 \times \cdots \times n_k
  • Multiply the number of available choices at each stage.
  • Apply restrictions before counting a later stage.
  • Check that the product counts the requested complete outcomes.

3. Set up the count

First, identify what the question asks you to count. Next, list the stages. For each stage, count the options that are available after applying any restrictions. Then multiply and state the result with a clear unit, such as outfits, arrangements, or lunch combinations.
For a three-stage task with four choices, three choices, and two choices, a slot display records one count for each stage. Multiplying those counts gives the number of complete outcomes.
When choices must be different, the later count may be smaller. If you choose a first book and then a different book for a second position from a group of seven, there are seven options for the first position and six for the second. Since the positions are distinct, switching the books gives a different ordered arrangement.
  • Translate each condition into a count of available choices.
  • For choices without reuse, update the later count.
  • Use words to say what one counted outcome represents.

4. Practice your reasoning

Before you calculate, ask: What are the stages? How many choices are available at each stage? Does an earlier choice remove or restrict a later option? What does one complete outcome look like?
A useful check is to make sure every factor matches one required stage. If a restriction reduces the available choices, the product should show that reduced count. Finally, compare your answer with the question: does it count complete outcomes of the kind requested?
  • Each factor should represent one stage.
  • A correct calculation also needs a correct interpretation.

Worked example

Build an outfit

A student has 4 shirts, 3 pairs of pants, and 2 jackets. Each shirt can be worn with any pair of pants and either jacket. How many outfits can be made by choosing one item from each category?
Outfit choice stages
Outfit choice stagesOne choice is made at each stage.Shirt4×Pants3×Jacket24 × 3 × 2 = 24

One choice is made at each stage.

  1. Name the stages
    There are three stages: choose a shirt, choose pants, and choose a jacket. The problem allows every shirt to be paired with every listed pair of pants and jacket.
  2. Record the choices
    There are 4 choices for the shirt, 3 for the pants, and 2 for the jacket. Each count represents one stage.
    4×3×24 \times 3 \times 2
  3. Multiply and interpret
    The product counts complete outfits, with one item chosen from each category.
    4×3×2=244 \times 3 \times 2=24
Answer: There are 24 different outfits.
Check: Each outfit is specified by one shirt, one pair of pants, and one jacket. All three required choices are counted.

Worked example

Choose different books in order

A display has 7 different books. A librarian chooses one book for the first position and a different book for the second position. How many ordered arrangements are possible?
Ordered book positions
Ordered book positionsThere are 7 choices for the first position and 6 for the second.Order matters7×67P2 = 42

There are 7 choices for the first position and 6 for the second.

  1. Count the first position
    Any of the 7 books can go in the first position.
    77
  2. Update the second count
    The first book cannot be used again, so 6 different books remain for the second position.
    66
  3. Multiply and interpret
    The positions are distinct. Switching the books changes the arrangement, so the count includes each possible first-and-second order.
    7×6=427 \times 6=42
Answer: There are 42 ordered arrangements.
Check: There are 7 choices for the first position and 6 for the second. The restriction against reusing a book is included.

Worked example

Select a lunch combination

A café offers 3 sandwiches, 4 fruit choices, and 3 drinks. A lunch deal includes one sandwich, one fruit, and one drink. How many different lunch combinations are available?
Lunch choice stages
Lunch choice stagesThe deal has three choice stages.Sandwich3×Fruit4×Drink33 × 4 × 3 = 36

The deal has three choice stages.

  1. Identify the stages
    A lunch deal requires three choices: one sandwich, one fruit, and one drink.
  2. Count the options
    There are 3 sandwich choices, 4 fruit choices, and 3 drink choices. The problem gives no restriction on how the choices can be paired.
    3×4×33 \times 4 \times 3
  3. Find the total
    Multiplying counts every complete lunch deal with one selection from each category.
    3×4×3=363 \times 4 \times 3=36
Answer: The café offers 36 different lunch combinations.
Check: Each counted combination contains exactly one sandwich, one fruit, and one drink.

Common mistakes and how to avoid them

Adding the numbers of choices at different stages.
Correction: When a complete outcome needs one choice from every stage, multiply the stage counts.
Using the original number of options at every stage even when an item cannot be reused.
Correction: Recount the options after applying the restriction. In the book example, the second position has 6 choices, not 7.
Counting only one part of the complete outcome.
Correction: List every required stage and include one factor for each stage.
Treating two ordered positions as the same outcome when their contents are switched.
Correction: If the positions are distinct, switching the choices gives a different arrangement.

Lesson summary

  • Break a multi-step task into stages.
  • Count the choices available at each stage, respecting restrictions.
  • Multiply the stage counts to find the number of complete outcomes.
  • State what the total represents and check that it matches the question.

Check your understanding

Question 1

A school event offers 5 activities and 2 snack choices. A student chooses one activity and one snack. How many activity-and-snack combinations are possible?
  1. 7
  2. 10
  3. 12
  4. 25
Show answer and explanation
10
There are 5 choices for the activity and 2 for the snack. Multiplying the stage counts gives 10 combinations.

Question 2

A player selects two different cards in order from 6 cards. How many ordered selections are possible?
  1. 12
  2. 30
  3. 36
  4. 15
Show answer and explanation
30
There are 6 choices for the first card and 5 remaining choices for the second. Their product is 30.

Question 3

A travel kit has 3 toothbrush options, 2 toothpaste options, and 4 soap options. How many kits can be made by choosing one of each?
  1. 9
  2. 12
  3. 24
  4. 36
Show answer and explanation
24
Multiply the choices for the three stages: 3 times 2 times 4 equals 24.

Key terms

Stage
One decision within a multi-step choice.
Outcome
One result of a choice; a complete outcome includes every required choice.
Fundamental counting principle
The rule that counts complete outcomes by multiplying the number of available choices at each stage.

Continue through Math 30-2

View the complete Math 30-2 Alberta Grade 12 Mathematics curriculum and lessons

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Alberta Grade 12 Mathematics (Math 30-2), expectation P4. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

Report a correction or ask a question