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Fundamental Counting Principle Calculator

Created by Maciej Kowalski, PhD candidate
Reviewed by Steven Wooding
Last updated: Jan 18, 2024

Welcome to Omni's fundamental counting principle calculator, where we'll cover one of the basic counting rules for possible outcomes of several choices. In essence, it says that we can multiply (that's why we often call it the multiplication principle) the number of options for one thing by the number of options for the other to obtain how many results we can find for the pair. Don't worry if the previous sentence sounds confusing; we'll see some real-life fundamental counting principle examples soon enough.

So what is the fundamental counting principle exactly?

What is the fundamental counting principle?

Say that you have a new purchase in mind and are not yet sure which option to choose. You need to pick the company, the model, and the color. If every company has the same number of models to choose from, and every model has the same number of possible colors, then the fundamental counting principle is what we turn to.

Formally speaking, the fundamental counting principle (aka the multiplication principle) is one of the basic counting rules in mathematics that helps us compute the number of specific objects. Let's let the principle speak for itself.

💡 The fundamental counting principle says that if you have sets A and B with a and b elements, respectively, then there are a * b distinct pairs (x,y) with x from A and y from B.

It's worth mentioning that the multiplication principle extends to more than pairs. After all, using the notation from the box above, if we add a set C with c elements, then we can count triples (x,y,z) from A, B, and C, respectively, by saying that we have a * b choices for the pair (x,y) and c choices for z. All in all, this gives (a * b) * c = a * b * c possible triples.

Alright, we might have seen the formal side to it, but what is the fundamental counting principle when applied to real-life scenarios? Well, let's explain it in a few examples.

There are other counting methods, like the factorial, combinations, and permutations. You can read more about these at the factorial calculator, combination calculator, and the permutation and combination calculator. Take a look!

Fundamental counting principle examples

Suppose that you'd like to order pizza. You feel picky, so you decide to browse through all the options you have on offer. How many are there?

Suppose that there are 4 pizza places around you, and each offers its products in 3 different sizes. Coincidentally, each restaurant has 12 different pizzas to choose from and 4 side sauces.

Let's use the counting rules, i.e., the fundamental counting principle to see how many combinations we have here. In principle, we're choosing:

  • The place;
  • The size;
  • The pizza itself; and
  • The side sauce.

According to the data above, we have 4 choices for the pizza place, 3 possible sizes, 12 different sets of toppings, and 4 sauces to choose from. Note how it's important each place had the same number of sizes, pizzas, and sauces on offer. Therefore, if we apply the multiplication principle to our problem, we'll see that we have:

4 * 3 * 12 * 4 = 576

options. Now that's what we call diversity!

Observe how we can apply the same reasoning to many other life-like problems, e.g., buying a car (the company, the model, the color), choosing a movie for the evening (the platform, the genre), etc. We can apply to them the same counting rules as in the fundamental counting principle above, as long as for each variant of the first thing, we have the same number of options for the second thing, and so on.

Lastly, before we let you go for today, let's see how to use Omni's fundamental counting principle calculator to solve all such problems in the blink of an eye!

Using the fundamental counting principle calculator

When you look at the fundamental counting principle calculator, you'll see two variable fields for the number of choices for the first and second things. The things in question can be pizza toppings, the color of a car, the score on dice when you roll it, or anything else of that sort. Check out the dice roller calculator to learn how to estimate the dice roll probability. However, every variant of the first characteristic must have the same number of options for the second (for instance, every car company must have the same number of colors available).

Although you first see only two variable fields, more will appear once you begin inputting data. In total, the fundamental counting principle calculator allows up to ten characteristics for the choice you're making! However, observe that the tool will already begin its calculations when you input two numbers. If you add new ones, the formula, and result will change accordingly.

That's all, folks! The fundamental counting principle calculator really is that simple, so make sure to play around with it and see how many options the world around you offers. Maybe we could calculate all possible pancake servings? Thick or thin. With jam, with syrup, with cream. And from there, it's different pancakes every breakfast!


How do I use the fundamental counting principle?

To use the fundamental counting principle, you need to:

  1. Specify the number of choices for the first step.
  2. Repeat for all subsequent steps.
  3. Make sure the number of options at each step agrees for all choices.
  4. Multiply the number of choices at step 1, at step 2, etc.
  5. The result is the total number of choices you have.

Can permutations be solved by using the fundamental counting principle?

Yes. For instance, if you have 3 objects that you want to put in order (say, an apple, an orange, and a lemon), then the number of possibilities is the number of permutations of a 3-element set, i.e., 3! = 3 × 2 × 1 = 6. Alternatively, we can use the fundamental counting principle to say that we have 3 choices for the first fruit, 2 choices for the second (after all, we've already chosen one to be the first), and 1 choice for the third (because we've already put the other two before). The fundamental counting principle then gives that, all in all, we have 3 × 2 × 1 = 6 choices.

Does order matter in the fundamental counting principle?

Yes. In the fundamental counting principle, we're numbering the steps in which we operate. For instance, if we're choosing ingredients for a pizza, but the first one we pick is the main one (i.e., there's more of it), then choosing pepperoni and onion gives a different one than picking onion and pepperoni: the second variant is more vegetarian than the first.

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

Let's talk pizzas. Suppose we're choosing three toppings for a pizza, but the first will be the main one, and the other two will be secondary (there'll be less of it) and tertiary (even less), respectively.

  • The fundamental counting principle counts how many choices we have for the main, secondary, and tertiary ingredients (order matters). Also, if we're feeling like it, we can allow the ingredients to repeat and have, for example, a super cheesy margherita with triple cheese.
  • A permutation counts ordered triples of ingredients, but it doesn't allow them to repeat, so no margherita this time.
  • A combination chooses three toppings and treats them equally, i.e., the order doesn't matter, and there's no main, secondary, or tertiary ingredient: there's the same amount of each. Usually, we count combinations without repetitions, but there are exceptions to this rule.

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

Before you use the fundamental counting principle, you need to:

  1. Make sure that the order of your choices matters.
  2. Calculate the number of options for the first step.
  3. Calculate the number of options for the second step.
  4. Make sure the number in point 3 doesn't depend on that from point 2.
  5. Repeat the reasoning for all consecutive steps.
  6. Unless the numbers don't agree at some point, you can use the principle.
Maciej Kowalski, PhD candidate
Number of choices for the 1st thing
Number of choices for the 2nd thing
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