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

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 \( n \times m \).

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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?

Solution:

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

                                                    

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?

Solution:

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.

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?

Solution

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?

Solution

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.