Fundamental Counting Principle Calculator

Fundamental counting principle calculator helps you determine the total number of possible outcomes for a series of independent events. It simplifies the process of counting and calculating the possibilities for various situations including permutations, combinations, and probability calculations.

How to use this fundamental counting principle calculator?

Enter the number of possible outcomes for the first, second, third option, and so on.

Press the Calculate button.

The calculator will provide you with the exact number of possible outcomes that you can have in a given scenario.

What is the fundamental counting principle?

The fundamental counting principle (FCP) refers to a basic rule in probability and combinatorics that is used to calculate the total number of possible outcomes in a situation.

It states that if there are n number of ways to do one thing and after that m number of ways to do another thing, the total number of ways to do could be calculated by multiplying both n and m (n × m).

Fundamental counting principle formula

The formula for the fundamental counting principle is:

$$ \text{Total Ways} = n_1 \times n_2 \times n_3 \times \ldots \times n_k $$

Where,

• n1, n2, n3,...,nk are the number of possible outcomes for that event

Learn how to use the fundamental counting principle practically by following the example below.

Example

let's consider a scenario where you want to choose a new outfit, and you have the following choices:

• Choose a Shirt: You have 4 different shirts to choose from (let \(n_1=4\))
• Choose Pants: You also have 3 different pairs of pants (let \(n_2=3\))
• Choose Shoes: Finally, you have 2 pairs of shoes to choose from (let \(n_3=2\))

Now, you want to calculate how many combinations you have to make an outfit:

Let’s find out using the fundamental counting principle formula:

\[ \text{Total Ways} = n_1 \times n_2 \times n_3 \]

Now, add the values into the equation:

\[ \text{Total Ways} = 4 \times 3 \times 2 = 24 \]

So, there are 24 different ways you can make an outfit by selecting one shirt, pair of pants, and pair of shoes.

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

FCP: Counts how many ways independent events can happen together, order doesn't matter here. Think about choosing toppings for ice cream, order doesn't affect the flavor!

Permutations: Counts how many ways to arrange distinct things in a specific order, order matters here! Think of different starting lineups for a baseball team.

Can permutations be solved by using the fundamental counting principle?

Yes, permutations can be solved using the fundamental counting principle, as the permutation formula is derived by considering the multiplication of choices at each stage in the fundamental counting principle.

However, it's important to note that the reverse is not always applicable. Some problems using the fundamental counting principle which allows for repeated choices, cannot be solved using the permutation formula.