The Fundamental Counting Principle Revision Notes for Grade 12 NSC Matric Mathematics

The Fundamental Counting Principle

Introduction

When we need to count large numbers of possible outcomes, simple methods like listing all possibilities become impractical. The fundamental counting principle provides a systematic mathematical approach to determine the total number of possible outcomes when multiple events occur together.

This principle is essential in probability theory because it allows us to calculate the size of sample spaces without having to enumerate every single possibility.

Definition and basic formula

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The fundamental counting principle states that when you have multiple independent events occurring in sequence, you multiply the number of possible outcomes for each event to find the total number of possible outcomes.

Definition: If there are $n(A)$ possible outcomes for event A and $n(B)$ possible outcomes for event B, then there are $n(A) \times n(B)$ possible outcomes when both events A and B occur together.

Formula for multiple events

For any number of independent events, the principle can be extended:

Two events: Total outcomes = $n_1 \times n_2$
Multiple events: Total outcomes = $n_1 \times n_2 \times n_3 \times \ldots \times n_k$

where $n_1, n_2, n_3, \ldots, n_k$ represent the number of possible outcomes for each respective event.

Choices without repetition

When different events have different numbers of possible outcomes, we multiply these numbers together. This commonly occurs when making selections from different categories.

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Worked Example: Meal Combinations

Question: A restaurant offers a lunch special with four components. Calculate the total number of different meals possible.

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Solution:

Step 1: Identify the number of events There are 4 meal components: sandwich, soup, dessert, and drink.

Step 2: Count the choices for each component

Sandwiches: 4 options
Soups: 3 options
Desserts: 2 options
Drinks: 5 options

Step 3: Apply the fundamental counting principle

Total possible meals = $4 \times 3 \times 2 \times 5 = 120$

Therefore, there are 120 different possible meal combinations.

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Worked Example: Dice and Coin Combination

Question: What is the total number of possible outcomes when rolling a die and tossing a coin?

Solution:

Step 1: Identify the events

Event A: Rolling a die (outcomes: 1, 2, 3, 4, 5, 6)
Event B: Tossing a coin (outcomes: heads, tails)

Step 2: Count the outcomes

Die: 6 possible outcomes
Coin: 2 possible outcomes

Step 3: Apply the principle

Total outcomes = $6 \times 2 = 12$ possible combinations

The sample space contains: (1,H), (1,T), (2,H), (2,T), (3,H), (3,T), (4,H), (4,T), (5,H), (5,T), (6,H), (6,T)

Choices with repetition

When the same set of choices is available for multiple events, we use the formula $n^r$ , where $n$ is the number of choices and $r$ is the number of times the choice is made.

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Formula for Repetition: When you have $n$ objects to choose from and you choose $r$ times (with repetition allowed), the total number of possibilities is:

$n \times n \times n \times \ldots \times n \text{ (r times)} = n^r$

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Worked Example: Coin Flips

Question: If a coin is flipped three times, how many different sequences of outcomes are possible?

Solution:

Step 1: Identify the repeated event

Each coin flip has 2 possible outcomes: heads (H) or tails (T).

Step 2: Count the number of repetitions

The coin is flipped 3 times.

Step 3: Apply the formula

Total outcomes = $2^3 = 2 \times 2 \times 2 = 8$

The tree diagram confirms there are 8 possible sequences: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

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Worked Example: Letter Arrangements

Question: Using the letters A, B, C, and D, how many three-letter patterns can be formed if repetition is allowed?

Solution:

Step 1: Identify the available choices

There are 4 letters: A, B, C, D.

Step 2: Count the positions

We need to fill 3 positions in the pattern.

Step 3: Apply the formula

Since repetition is allowed, each position can be filled with any of the 4 letters.

Total patterns = $4^3 = 4 \times 4 \times 4 = 64$

Examples include: AAA, ABC, BAD, DCA, etc.

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Worked Example: Sports Results

Question: A school plays 6 football matches. Each match can result in a win, draw, or loss. How many different sequences of results are possible?

Solution:

Step 1: Identify the outcomes per event

Each match has 3 possible outcomes: win, draw, or loss.

Step 2: Count the number of events

There are 6 matches.

Step 3: Apply the formula

Total sequences = $3^6 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 729$

Tree diagrams vs counting principle

Tree diagrams are useful visual tools for small numbers of outcomes, but they become impractical as the number of events increases.

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When to use tree diagrams:

Few events (typically 3 or fewer)
Need to see all possible outcomes clearly
Working with conditional probability

When to use the counting principle:

Large numbers of outcomes
Quick calculation needed
Multiple independent events
Exam situations requiring efficient solutions

Example: For 6 coin flips, a tree diagram would have $2^6 = 64$ endpoints, making it extremely complex to draw. The counting principle gives the answer immediately: $2^6 = 64$ .

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Critical Exam Points:

Order matters: The fundamental counting principle assumes that the order of events is significant.
Independence: The events must be independent - the outcome of one event doesn't affect the outcomes available for other events.
Systematic approach: Always identify the number of events first, then count outcomes for each event, then multiply.
Check your work: Ensure you've correctly identified whether repetition is allowed or not.

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Key Points to Remember:

The fundamental counting principle states: multiply the number of possible outcomes for each independent event to find the total outcomes
For different choices: multiply the number of options for each event $(n_1 \times n_2 \times n_3 \times \ldots)$
For repeated choices: use the power formula $n^r$ where $n$ = number of choices and $r$ = number of repetitions
Tree diagrams become impractical with many events - use the counting principle for efficiency
Always check whether repetition is allowed before choosing your approach

The Fundamental Counting Principle (Grade 12 NSC Matric Mathematics): Revision Notes