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Fundamental Counting Principle

Fundamental Counting Principle

In this mini-lesson, we will explore the fundamental counting principle by learning about the fundamental counting principle meaning, using the  fundamental counting principle examples while discovering the interesting facts around them.

Here's a fundamental counting principle calculator. Enter the values and check how this principle works.

Lesson Plan

What Is the Fundamental Counting Principle?

The fundamental counting principle is a rule to count all the possible ways for an event to happen or the total number of possible outcomes in a situation.

It states that when there are \( n \)  ways to do one thing, and \( m \) ways to do another thing, then the number of ways to do both the things can be obtained by taking their product. This is expressed as \( n \times m \).

How to Use the Fundamental Counting Principle to Determine Sample Space?

Let's get acquainted with sample space first.

Sample space is referred to as the collection of all possible outcomes in any random experiment.

Now, that we know what is sample space, let's see the use of the fundamental counting principle to determine sample space.

Let's see a few fundamental counting principle examples to understand this concept better.

Example 1:

Claire has \(2\) shirts and \(2\) skirts of different colors in her closet. The colors of the shirts are pink and black, while the colors of the skirt are black and white. She wore one of the combinations, which were a pink shirt and a white skirt. What do you think are the other possible combinations she could try?

Fundamental Counting Principle: Example


She could wear any of the following \(3\) combinations apart from the one which is worn by her.

                             Fundamental Counting Principle: Example                       

This means using the 2 shirts and 2 skirts, she could dress up in 4 different ways.

If we the apply the fundamental counting principle, we get to know that she can wear either the pink shirt or the black shirt. Similarly, she can either wear the white skirt or the black skirt.

Hence, she can wear her shirts in 2 ways and skirts in 2 ways. To get the total number of combinations possible, we take the product of 'the number of ways in which she can wear her shirt' with 'the number of ways she can wear her skirt'. Hence, the number of combinations are represented as \( 2 \times 2 = 4 \). This shows that the sample space is 4

Example 2:

Brad has 2 bananas, 3 apples and 3 oranges in a basket. In how many different ways can he consume the fruits in the basket?

Fundamental Counting Principle: Example


Applying the above rules, we know that the total number of ways in which he can consume the fruits in the basket  is:  \( 2 \times 3 \times 3 = 18 \).

Hence, he has 18 different ways to consume the fruits. This tells us that 18 is the sample space in this case.

Any choice made by him will be within these 18 ways.

important notes to remember
Important Notes
  • It is possible to find the sample space without writing it down, using the fundamental counting principle.
  • The fundamental counting principle is also used when the sample space is very large. This eases the trouble of writing it down. It can be used in probability as well.

Solved Examples

Example 1



Wendy went to buy an ice cream from a seller who sells 3 different flavors of ice creams, vanilla, chocolate and strawberry and he gives 6 different choices for cones. How many choices does she have?

Solved Example: Fundamental Counting Principle showing example of ice cream cones


The ice cream seller sells 3 flavors of ice creams, vanilla, chocolate and strawberry giving his customers 6 different choices of cones.

Wendy has 3 choices for the ice cream flavors and 6 choices for the ice cream cones.

Hence, by the fundamental counting principle, the number of choices that Wendy can be given are \( 3 \times 6 = 18 \).

Wendy can choose from any of the 18 possible combinations.

\(\therefore\) Wendy has 18 choices.
Example 2



Ashton knows there are 7 daily newspapers and 4 weekly magazines published in his town. If he wants to subscribe to one daily newspaper and one weekly magazine, how many choices does he have?

Solved Example: Fundamental counting principle


Ashton knows there are 7 daily newspapers and 4 weekly magazines published in his town.

He will apply the fundamental counting principle to calculate the choices that he has.

Hence, Ashton will take the product of "the number of ways in which he can select a daily newspaper" and "the number of ways in which he can select a weekly magazine".

The number of choices will be calculated as \( 7 \times 4 = 28 \).

\(\therefore\) Ashton has 28 choices.
Example 3



There are 6 children in a classroom and 6 benches for them to sit. The class teacher makes them sit at a different place every month. In how many ways can she make them sit in the classroom?

Solved Example: Fundamental counting principle


There are 6 children and 6 benches for them to sit.

Hence, their teacher will apply the fundamental counting principle to find the number of ways in which she can make them sit.

The number of ways in which she can make the children sit in the classroom is \( 6 \times 6 = 36 \).

\(\therefore\) There are 36 ways.

Interactive Questions

Here are a few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result.

Challenge your math skills
Challenging Questions
  1. Lily has 4 shirts, 3 pants, 2 pairs of shoes and 3 handbags. In how many ways can she choose her outfits such that she selects 1 shirt, 1 pant, 1 pair of shoes and 1 handbag ?
  2. Christopher went to a library to get a novel and a history book. There are 60 novels and 148 history books in the library. How many combinations of books can he choose? 

Let's Summarize

The mini-lesson targeted the fascinating concept of fundamental counting principle. The math journey around fundamental counting principle starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

Frequently Asked Questions (FAQs)

1. What is the difference between fundamental counting principle and permutation?

In a problem, where the objects can be repeated, permutation cannot be used. In such cases the fundamental counting principle is used.

2. What is r in the combination formula?

In the combination formula,"r" represents the number of items that can be chosen at once.

3. Why is the fundamental counting principle important?

The fundamental counting principle can be used for problems having large sample spaces, problems having more than two choices and can also be applied in probability.

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