What Are Combinatorics in Mathematics?

Combinatorics tells you how many different ways something can be combined.

There are several different formulas in the theory of combinatorics. That’s why it’s smart to familiarize yourself with what combinatorics is all about.

Imagine that you one day can’t decide what to wear among three t-shirts, two pairs of pants, and two sweaters. In how many different ways can you choose one sweater, one t-shirt, and one pair of pants? Combinatorics can help answer this type of question.

When you’re calculating how many separate ways something can be combined, you have to multiply the number of possibilities with each other. In this case it’ll be:

There are 12 separate combinations of clothes to put on in the morning.

Example 1

You’re out to dinner at a restaurant, and look at the menu. You don’t like appetizers, but you’re very fond of dessert. For that reason, you will eat one main course and one dessert. There are three different main courses: Sausages, spaghetti and pizza. You can choose from three different desserts: Brownies, sweet buns or cake. How many different two-course dinners can you order?

The solution will be to multiply the number of dishes in each category with each other: The number of main courses multiplied with the number of desserts.

The number of possible combinations then becomes $3\cdot 3=9$. That means there are nine different ways you can combine one main course with one dessert when there are three main courses and three desserts.

Rule

Combinatorics

When solving combinatorial tasks, multiply the number of items in each category with each other.

You need to create a four-digit code on a mobile phone. How many different codes can you create?

To be able to calculate how many different codes there are, You must first find out how many options you have for each digit. For the first digit in the code you have 10 options, the digits from $0$$9$. The same applies to digit number two, three and four in the code. So there are 10 different options for each of the digits.

Then you need to multiply the number of possibilities for each digit in the code with each other, giving you

 $10\cdot 10\cdot 10\cdot 10=10\phantom{\rule{0.17em}{0ex}}000.$

There are actually $10\phantom{\rule{0.17em}{0ex}}000$ different options for a four-digit code! Imagine how many people have the same code.

One is often interested in the number of ways in which something can be done. To talk more about combinatorics, there are a couple of terms it’s important to know:

Theory

ImportantTerms

Ordered set

means that the order matters. This means that, for example, the sequences 1, 2, 3, 4 and 4, 3, 2, 1 counts as two different outcomes.

Unordered set

means that the order does not matter. This means that, for example, the sequences 1, 2, 3, 4 and 4, 3, 2, 1 are the same outcome.

Replacement

means that the same outcome can happen several times.

No replacement

means that the each outcome can happen only once.

Example 2

Mickey Mouse, Goofy, Donald Duck, and Pluto are four good friends who are going to make dinner together, but they forgot to go grocery shopping. That means they’ll have to send two of them to the grocery store. Which two they send is an unordered set (because there’s no difference between sending Mickey and Goofy, and sending Goofy and Mickey). Since they have to send two separate people you can view this as an experiment with no replacement.

Later in the evening the mood is getting good, and the four friends decide to play Scrabble. When the game is finished, one of them will be the winner, one will be in second place, one in third place, and one in last place. This ranging is an ordered set because the order matters—you would rather win than lose. It is also a set with no replacement, because the same person cannot come in first and third place at the same time. An ordered set with no replacement is also knows as a permutation.