The Fundamental Principle of Counting

The Fundamental Principle of Counting is a simple rule that helps us figure out how many different ways we can do something when there are choices involved.

Simple Rule: If you have a number of choices for one thing and a number of choices for another thing, you can find the total number of possible outcomes by multiplying the number of choices together.

For example, if you can choose between $3$ different shirts and $2$ different trousers, you can find the total number of outfits by multiplying the choices: $3 \times 2 = 6$ .

Formal Rule: If one event has $m$ possible outcomes, and another event has $n$ possible outcomes, then the total number of possible outcomes for both events happening together is $m × n$ .

Now, let's see how this works in real-life situations!

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Example 1: Choosing an Outfit Let's say you have:

3 shirts (Red, Blue, Green)
2 trousers (Jeans, Black trousers)
2 pairs of shoes (Trainers, Boots) To find out how many different outfits you can make, multiply the number of choices: $3 \times 2 \times 2 = 12$ So, you can make 12 different outfits.

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Example 2: Making a Meal Imagine you're at a restaurant, and you want to order a meal. You can choose:

2 starters (Soup, Salad)
3 main courses (Pizza, Pasta, Burger) How many different meals can you make? Multiply the choices: $2 \times 3 = 6$ So, you can create 6 different meal combinations.

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Example 3: Planning a Day Out You're planning a day out, and you can choose:

4 different activities (Cinema, Bowling, Park, Swimming)
3 places for lunch (Café, Fast Food, Restaurant)
2 options for dessert (Ice cream, Cake) To find out how many different day-out plans you can make, multiply the choices: $4 \times 3 \times 2 = 24$ So, you can create 24 different combinations for your day out.

Why is this Important?

This principle helps you solve problems quickly, especially when you need to count the number of possible outcomes. Whether you're making combinations of clothes, food, or planning a day out, understanding this rule allows you to work out the total possibilities with ease.

Example Problem

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Question: A travel company offers different vacation packages. You can choose from:

3 destinations (Paris, Rome, New York)
2 types of accommodation (Hotel, Apartment)
3 activities (City Tour, Museum Visit, Beach Day)
2 types of transport (Plane, Train) How many different vacation packages can you create?

Step 1: Write down the number of choices for each option:

Destinations: 3 choices
Accommodation: 2 choices
Activities: 3 choices
Transport: 2 choices Step 2: Multiply the choices together: $3 \times 2 \times 3 \times 2 = 36$ Answer: There are 36 different vacation packages you can create.

Break it down like this:

For each destination, you can choose between hotel or apartment, then pick one activity, and finally decide on plane or train transport.

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Exam Tips:

Draw it out: If you find it tricky, draw pictures or make a list to help you see all the choices.
Check your steps: Make sure you include all the parts of the question before multiplying.
Practice makes perfect: The more you practice, the easier this will become. Try practicing with different examples, like choosing outfits, making meals, or planning trips.

The Fundamental Principle of Counting Simplified Revision Notes for Junior Cycle Mathematics

The Fundamental Principle of Counting

Why is this Important?

Example Problem

500K+ Students Use These Powerful Tools to Master The Fundamental Principle of Counting For their Junior Cycle Exams.

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