What is combinatorics?

Photo by Waldemar Brandt on Unsplash

This is the first in a series of articles focused on combinatorics. When I first took a discrete math course in college, I jokingly told my friends that here I was, a math major, learning how to count. Honestly though, I don’t know of a better way to describe combinatorics. I can say that it’s one of my favorite fields of math (along with graph theory) not just because the subject itself is so intriguing, but because it’s something you can easily talk about with people who aren’t as comfy with mathematics yet. It’s simultaneously accessible and rich with open, unanswered questions. Before we dive into the nitty gritty though, let’s zoom out for a sec and see where combinatorics stands in the grand scheme of mathematics.

Let’s start with a broad overview. Here’s a really cool video made by Dominic Walliman that illustrates the various fields of math and how they fit together.

In his video, Dominic begins by dividing math into two main (although definitely not exclusive) categories: pure and applied. But that’s not the only way to see math. We could also start by considering two very different categories: discrete and continuous. Discrete math deals with objects that are distinct from one another and can be counted with the natural numbers (0,1,2,3,…). This means we don’t really bother thinking about real numbers, calculus, Euclidean geometry, any of that messy stuff (these are all continuous). There are many fields that exist under the broad umbrella of discrete math. Some of these include graph theory, game theory, number theory, computer science, and of course, combinatorics. Broadly speaking, combinatorics is the math of counting things. To get a sense of what that actually means, here are a few questions that combinatorics can help answer:

  • Given a standard deck of 52 cards, how many cards would you need to draw in order to guarantee you have a spade in your hand?
  • How many distinct passwords can you create with the letters A-Z and numbers 0–9?
  • How many ways can we distribute 20 identical pencils to 7 students?
  • How many distinct ways can we rearrange the letters in MISSISSIPPI?

These are just a few examples of questions we’re going to work through in this series. Along the way, we’ll also see some cool families of numbers, some intriguing applications, and a heck of a lot of cool math.
You ready? Let’s goooo!

If your interest is piqued and you’re already ready for more, go check out this cool video from Numberphile.

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