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PC1 · Apply the fundamental counting principle

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

Alberta Grade 12 Mathematics

Permutations, Combinations, and Binomial Theorem

Applying the fundamental counting principle

Many tasks involve making a sequence of choices. You might choose one item from each of several categories, or fill several positions in a code. The fundamental counting principle gives a direct way to count the possible complete results. The main work is to identify the stages and count the choices available at each stage.

What you will learn

  • Identify the stages in a task.
  • Count the available choices at each stage, including when the number changes.
  • Multiply stage counts to find the number of complete outcomes.
  • Check that the count follows every condition in the task.

Review: stages and outcomes

A stage is one part of a task that must be completed. Choosing an outfit with one shirt and one pair of pants has two stages: choose a shirt, then choose pants.
An outcome is one complete result. A particular shirt paired with particular pants is one outfit, so it is one outcome. A choice at only one stage is not a complete outfit.
Before calculating, identify what must be chosen and read the conditions. Ask whether making one choice changes the choices available at a later stage.
  • A stage is one part of a task.
  • An outcome is one complete result.
  • Count only choices that meet the stated conditions.

The fundamental counting principle

The fundamental counting principle says that when a task has stages, multiply the number of available choices at each stage to find the number of complete outcomes.
Suppose a first stage has four choices and a second stage has three. Each first choice can be paired with each second choice, giving twelve complete results. For more stages, continue multiplying the choice count for every stage.
The number of choices may change from one stage to the next. If a choice cannot be repeated, making an earlier choice can remove an option later. Count the choices that remain at each stage, then multiply those counts.
A set of slots can show the stages. Write the available choice count for each slot. The slots represent the positions or stages in the task, not specific outcomes.
N=c1c2⋯ckN=c_1c_2\cdots c_k
  • Multiply the available choices at every stage.
  • Recount later choices when earlier choices affect them.
  • The product counts complete outcomes.

A reliable method

First, describe the task as a sequence. A three-position code has three stages, one for each position. A meal with a main dish and a drink has two stages.
Next, count the available choices at each stage. Apply the conditions in the question. If an item cannot be repeated, account for that restriction in the later counts.
Then multiply the stage counts. Check that the answer counts complete results, not just choices for one stage. For a code, the answer should count complete codes.
A short list or table can help when the counts differ. Record each stage and its available choices before multiplying. This makes it easier to notice a restriction.
  • List the stages before calculating.
  • Record a choice count for each stage.
  • Check what complete result the product counts.

Practice and checking

Try a task with a restriction, such as forming a three-position code from digits when repetition is not allowed. Count the choices for the first position, then count what remains for the next positions.
To check your result, ask whether every counted result follows the rules and whether every valid result has been included. The principle applies when each complete result can be described by making one choice at every stage.
  • Apply each restriction where it affects the available choices.
  • Check that all valid complete results are counted.

Choices by stage

TaskStage countsTotal
Outfit4, 3, 224
Three-position code10, 9, 8720
Bus-route plan3, 412

Worked example

Choosing an outfit

A student can choose one of four shirts, one of three pairs of pants, and one of two pairs of shoes. How many outfits are possible?
Outfit choice stages
Outfit choice stagesEach slot represents one stage in choosing an outfit.Order matters33P1 = 3

Each slot represents one stage in choosing an outfit.

  1. Identify the stages
    There are three stages: choose a shirt, choose pants, and choose shoes. Each complete outfit uses one choice from each stage.
  2. Multiply the counts
    Choosing a shirt does not change the available pants or shoes. Multiply the three counts to find the number of complete outfits.
    4×3×2=244\times3\times2=24
Answer: There are 24 possible outfits.
Check: Each shirt can be paired with any of the three pants choices and either pair of shoes. The product counts every such outfit.

Worked example

A code with no repeated digits

A three-position code uses digits from 0 through 9, and no digit may be repeated. A code may begin with 0. How many codes are possible?
Code positions
Code positionsEach slot represents a code position. The available choices decrease because digits cannot repeat.Order matters10×9×810P3 = 720

Each slot represents a code position. The available choices decrease because digits cannot repeat.

  1. Count the first position
    All ten digits, including 0, are available for the first position because the question allows a leading 0.
    1010
  2. Count the remaining positions
    After one digit is used, nine digits remain for the second position. After two are used, eight remain for the third. Multiplying counts the complete codes.
    10×9×8=72010\times9\times8=720
Answer: There are 720 possible codes.
Check: The decreasing counts enforce the rule against repeated digits. The first position has ten choices because a leading 0 is allowed.

Worked example

Planning a two-part trip

A traveller can take one of three buses from a town to a station. From the station, the traveller can take one of four buses to a museum. How many bus-route plans are possible?
Two route stages
Two route stagesThe two slots represent the two bus choices in a complete route plan.Order matters22P1 = 2

The two slots represent the two bus choices in a complete route plan.

  1. Identify the stages
    The first stage is choosing a bus from the town to the station. The second is choosing a bus from the station to the museum.
  2. Multiply the counts
    Each of the three first bus choices can be followed by any of the four second bus choices. Multiplication counts every complete plan.
    3×4=123\times4=12
Answer: There are 12 possible bus-route plans.
Check: Each plan includes one bus choice for each part of the trip, so the product counts complete plans.

Common mistakes and how to avoid them

Adding the choices at each stage instead of multiplying them.
Correction: A complete result uses one choice from every stage. Multiply the stage counts to count complete results.
Using the same choice count at later stages when repetition is not allowed.
Correction: After a choice is used, recount the available choices for the next stage.
Excluding 0 from the first position of a code without being told to do so.
Correction: Follow the wording. If a leading 0 is allowed, count 0 as an option in the first position.

Lesson summary

  • Break the task into stages.
  • Count the available choices at every stage using the stated conditions.
  • Multiply the stage counts to find the number of complete outcomes.
  • Check that the product counts valid complete results.

Check your understanding

Question 1

A snack consists of one of five sandwiches and one of three drinks. How many different snacks are possible?
  1. 8
  2. 15
  3. 53
  4. 2
Show answer and explanation
15
There are two stages with five and three choices. Multiply to get 15.

Question 2

How many two-position codes can be made from digits 0 through 9 if digits cannot repeat and a leading 0 is allowed?
  1. 90
  2. 100
  3. 81
  4. 20
Show answer and explanation
90
There are ten choices for the first position and nine remaining choices for the second, so there are 90 codes.

Question 3

A three-stage task has 2 choices at the first stage, 6 at the second, and 3 at the third. How many complete outcomes are possible?
  1. 11
  2. 36
  3. 18
  4. 263
Show answer and explanation
36
Multiply the stage counts: 2 times 6 times 3 equals 36.

Key terms

Stage
One part of a task that must be completed.
Outcome
One complete result of a task.
Fundamental counting principle
The rule that the number of complete outcomes in a task with stages is the product of the available choices at each stage.

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Published by DoAssignment. This reviewed lesson follows Alberta Grade 12 Mathematics (Math 30-1), expectation PC1. It is a study resource, not an official curriculum publication.

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