Thanks for using our fundamental counting principle calculator; we hope you’ll find it helpful. Here, we will also give you a brief look into how this calculator works and explain some basics of the fundamental counting principle.

Even though the name fundamental counting principle itself sounds like something that came straight out of Einstein’s books, this is actually a very simple and handy math formula that can help you in everyday life situations.

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What is the fundamental counting principle?

It can get a bit more complicated as you add more variations (that’s why we made this calculator, so you don’t have to do it all manually). Still, in essence, the fundamental counting principle is a way to calculate the exact value of possible variation outcomes for anything.

We can use this principle whenever we need to determine the total number of outcomes in a situation with different variations.

Fundamental counting principle formula

There is no specific formula for the fundamental counting principle as it is essentially just the multiplication of all possible variations to get an exact number of outcomes. However, even though the formula is very simple, you might need to see some examples to understand it.

If we have an x number of variations combined with a y number of variations, to find the exact number of outcomes, we will multiply x with Y.

X*Y= number\hspace{1.5 mm} of \hspace{1.5 mm} outcomes

No matter how many variations there are to be combined, you can still apply the same formula:

X*Y*Z\left ( and\hspace{1.5 mm} so \hspace{1.5 mm} on \right )

As you can see, we are just multiplying the numbers of variations with each other. Thus, the fundamental counting principle is also called ‘the multiplication principle’, ‘the counting rule’, or ‘the basic counting principle’.

Fundamental counting principle examples

The best way to understand the fundamental counting principle is by applying it to some real-world problems.

1: Calculating the exact number of t-shirt variations to be printed out for a small t-shirt business

Let’s say that you start a small t-shirt shop business, and initially, you want to sell seven different t-shirt designs, and each of these t-shirt designs will be printed on five different sizes of t-shirts (S, M, L, X, and XL).

If you want to know how many t-shirt choices in total will be there once you print your first batch for all the given sizes and designs, you can calculate this quickly by applying the fundamental counting principle formula.

To get the exact number of t-shirt choices, you must multiply the number of designs you have with the number of t-shirt sizes you want to sell.

  • t-shirt design variations = 7
  • t-shirt size variations = 5
7*5=35

Therefore, seven t-shirt designs multiplied by five t-shirt sizes equals 35 different choices that you will have to print to get your first batch ready for sale.

Sounds like plenty of t-shirts to print out only to get the first batch done? Well, that’s why calculating everything beforehand will save you from unpleasant surprises in the future.

Now, let’s say you are a bit more ambitious and want to get highly agile with this business from the first day, and you want to let your customers also be able to choose between three different materials. Also, you don’t want to offer just simple unisex t-shirts, and you want your customers to be able to choose between male or female kinds of t-shirt designs.

To count of all the outcomes you will need for your first batch to be fully printed and ready for shipping, you can apply the same formula again:

  • Seven variations of t-shirt designs
  • Five variations of t-shirt sizes
  • Two variations of the material
  • Two variations regarding the gender
7 * 5 * 3 * 2 = 210

Seven different designs to be printed out times five different sizes times two different materials times two different designs (male/female) equals exactly 210 choices of t-shirts to be printed out.

2: Calculating the number of product variations for a small kebab shop

We will try to get it a bit more complicated in this example. Still, eventually, you will see that calculating the exact number of outcomes is easy if we apply the fundamental counting principle.

Let’s say you are running a small doner kebab shop, and you want to expand your business by listing your products on an online delivery app. So you reach out to the app manager to get it all sorted out, and they get back to you saying they need to know the exact number of kebab variations you will be offering online.

This can also be quickly calculated using the fundamental counting principle. Still, we need to make sure that we correctly count all of the variations that we will offer to the customers.

So, let’s say your kebab shop sells two kinds of meat – beef, and chicken. And you also want to include a vegetarian kebab in your online offer. Also, you want to offer two kinds of toppings for all of the kebabs (hot sauce and mild sauce), and you also want to offer a no topping option. In addition to that, you also offer just three kinds of vegetables that can be added as an extra onion, lettuce, and tomato. Furthermore, you also want to offer a no vegetable option. Finally, you want to sell your kebab in three sizes: small, medium, and large.

Now, let’s break our offer down and let’s count all variations that we have there (remember – correctly counting the variations is the most important part of this task):

  • Beef, chicken, and vegetarian – this gives us three possible variations regarding the meat (or no meat in the case of vegetarian kebab) included within each kebab.
  • Mild topping, hot topping, both toppings together, and no topping at all – there are four variations regarding the toppings (note how no topping option is counted as variation as well!)
  • Onion, lettuce, tomato, no vegetables at all – so far, there are four variations regarding the vegetables to be included. However, we must not forget that customers should also have an option to choose to have all of the vegetables together, onion and lettuce only, onion and tomato only, and lettuce and tomato only! Thus, we have eight variations in total here! (You can learn a better way to count this effortlessly in the section below where we explain combinations and permutations)
  • Small, medium, large – there are three variations in total regarding the size.

Now when we have all of the variations counted correctly, we can apply the fundamental counting principle to get the final number of all outcomes:

3 * 4 * 8 * 3 = 288\hspace{1.5 mm} outcomes \hspace{1.5 mm}in \hspace{1.5 mm}total!

Fundamental counting principle in statistics

The fundamental counting principle is widely used in statistics. As you can see from the examples above, it is hard to avoid using it when working with plenty of variations and possible outcomes.

This principle is also a backbone of combinatorics, a branch of mathematics that is used for calculations regarding counting and finding possible outcomes within finite structures. It is also used in combinations and permutations formulas that are also widely used in statistics.

Fundamental counting principle, combinations, and permutations

Okay, this is the part where the equations start to get weird symbols like exclamation marks! But don’t get discouraged with that, as you will see that these formulas we are going to explain here are still very simple as they essentially rely on the fundamental counting principle.

We need to know the difference between combinations and permutations to understand this.

Combinations are a group of all possible outcomes of a certain number of variations where the order does not matter.

On the other hand, permutations are a group of all possible outcomes but where some order needs to be followed.

The best way to see the difference is with real-world examples.

1. Combinations

Let’s say that you are a small business owner with a team of six employees in total, and you want to make shifts, so all of the employees get to work together at one point as a part of a team-building effort.

The working hours are divided into two shifts, and only three employees can be there at once.

This is a perfect example of combinations where the order does not matter, and you only need to make sure to eventually combine all of the employees to work with each other.

If we assign a number to every employee (1,2,3,4,5,6) we can have these outcomes:

[1, 2, 3], [1, 2, 4], [1, 2, 5], [1, 2, 6], [2, 3, 4],
[2, 3, 5], [2, 3, 6], [3, 4, 5], [3, 4, 6], [1, 3, 4],
[1, 3, 5], [1, 3, 6], [1, 4, 5], [1, 4, 6], [1, 5, 6],
[2, 4, 5], [2, 4, 6], [2, 5, 6], [3, 5, 6], [4, 5, 6]

If we count all of the outcomes, we can see that there are a total of 20 outcomes. Note that each combination is unique, and there are no repeating elements. This would be a permutation problem if we had a combination of repeating elements. We will explain this in detail below when describing the permutation example.

Combinations formula

What we did in here ‘manually’ can be calculated more easily by applying the combinations formula:

C is the number of combinations when we want to choose k number of elements from an n number of elements.

_{n}\textrm{C}_{k}=\binom{n}{k}=\frac{n!}{k!\left ( n-k \right )!}

! is a symbol for the factorial – to get factorial of a certain number, we need to multiply all of the consequent numbers, including the given number, all the way to 1.

Formula for calculating factorial of a n number would be:

n!=n\cdot \left ( n-1 \right )\cdot \left ( n-2 \right )\cdot \left ( n-3 \right )...\left ( 3 \right )\left ( 2 \right )\left ( 1 \right )

For example, factorial of number 6 would be:

6! = 6 * 5 * 4 * 3 * 2 * 1 = 720

So, if we apply this formula to the example above, we will have:

k = 3
n = 6

_{6}\textrm{C}_{3}=\binom{6}{3}=\frac{6!}{3!\left ( 6-3 \right )!}=20

Note the difference between the cases we need to apply the combinations/permutations formula instead of using the fundamental counting principle itself.

2. Permutations

Before moving to the permutations example, note that the factorial of a number is permutation itself, i.e., n! gives the number of permutations of n items!

We can take the same example with employees, but we will make it that way the order does matter. Thus, we will have repeating variations ( [3, 2, 1], [2, 3, 1] and [1, 2, 3] will be all different possibilities that we need to take into account, which was not the case with combinations).

So, let’s say that we have six employees again, and we want to know how many different ways they can be lined up in a group of three.

Note that the order here does matter as we will be lining all employees in a group of three.

To calculate this, we will apply the permutations formula.

Permutations formula

P is the number of ways to choose and arrange k elements from the group of n elements.

_{n}\textrm{P}_{k}=k!\binom{n}{k}=k!\cdot \frac{n!}{k!\left ( n-k \right )!}=\frac{n!}{\left ( n-k \right )!}

So, if we apply this formula to the example above, we will have:

k = 3
n = 6

_{6}\textrm{P}_{3}=k!\binom{6}{3}=3!\cdot \frac{6!}{3!\left ( 6-3 \right )!}=\frac{6!}{\left ( 6-3 \right )!}=120

How to use the fundamental counting principle calculator?

Our fundamental counting principle calculator is very simple to use.

You can use it to calculate as many choices as you need.

Note that initially, only three choices are shown on the screen. However, as you add them up, the calculator will show an extra space for the next choice on the screen.

You can see the result at the bottom of the screen instantly.

FAQ

Can permutations be solved by using the fundamental counting principle?

Whenever we have a problem requiring some counting to get the solution, we can apply the fundamental counting principle, permutations, or combinations formula.

Yes, permutations can be solved using the fundamental counting principle. Still, it is not always possible to do the opposite, i.e., to use the permutations formula to solve all fundamental counting principle problems (because in permutations, the order does matter, which is not the case with the fundamental counting principle).

For example, suppose we apply the fundamental counting principle to the permutations example above (where we needed to calculate how many rows of three can six different employees be lined up). In that case, we will get the same solution as if we apply the permutations formula:

6 * 5 * 4 = 120

There are only three ways that six employees can be lined up.

For the first place in the line, there are six possible variations. The second place in the line has five possible variations (as the first place has been taken already by one of the employees), and then for the third place in the line, we are left with four possible variations. Now when we know the exact count of possible variations for each place in the line, we can apply the fundamental counting rule and multiply these together to get the total count of outcomes.

Does order matter in the fundamental counting principle?

No, when calculating the number of outcomes in a certain situation, the order of variations is not important. For this reason, we can’t use the permutations formula to solve the fundamental counting principle problems, but we can do vice versa!

How are the fundamental counting principle, permutations, and combinations related?

Both the permutations and combinations formula heavily rely on the fundamental counting principle. Whenever we have a counting problem, we can use one of these three to solve it.

The combinations and permutations formulas are extended versions of the fundamental counting principle.

How do I know when to use the fundamental counting principle?

The fundamental counting principle can be used whenever we have a counting problem. For example, whenever we need to count all possible outcomes for different variations, we can use the counting principle to solve it.

Summary

The fundamental counting principle is an important math equation that helps us whenever we have a problem where we need to count certain variations and outcomes.

Our fundamental counting principle calculator can help you do this without having to calculate manually.

If you are a big math fan or need to do various calculations, please check out the other calculators available on our app.

If you found this calculator helpful, you might also want to check our Probability of 3 Events Calculator, Coin Flip Probability Calculator, Trigonometry Calculator.