You know what’s really fun? Counting. That’s what combinatorics is all about! So, count on enjoying this intro to combinatorics.

As the name would suggest, combinatorics focuses on the combinations of sets of data, whether this is some highly abstract data, particles, or every day objects like pencils. To make the counting of combinations more concrete, let’s work with an example:

Here’s 3 marbles with one being purple, one red, and one green. The question is how many ways can these three marbles be arranged. One arrangement is as the photo showed: Purple, Red, Green. Here’s them all:

Purple, Red, Green

Purple, Green, Red

Red, Purple, Green

Red, Green, Purple

Green, Purple, Red

Green, Red, Purple

That’s 6 total combinations. How were these combinations decided? Well, for the first marble, we have three choices since we have three colors. For the middle marble, only two of the colors remain, and the last marble must be the remaining color. Here’s the Purple, Green, Red example of this:

Now, if we multiply each number of options, we get 3*2*1 which is 6. Some people may recognize this as the factorial which is defined as so:

n! = n*(n-1)*(n-2)*…*1

The factorial counts the number of possible combinations in the exact manner that we did with the marbles. This is a ton easier than having to write out every combination. For example, having 4 marbles would require 24 combinations because 4! = 4*3*2*1 = 24.

 

Now that we can count combinations, let’s use them. We have 6 indistinguishable marbles and must arrange them into the two sides.

Because we cannot tell any 2 marbles apart, we find repeated combinations. For example. if I want 2 marbles on the left and 4 on the right, I could have the first and second marbles on the left, the first and last marbles, and so on for a bunch of combinations. Using factorials, here’s the formula for counting the number of ways to get a set up:

n!/[k!*(n-k)!]

For us, n is 6, and k will be any value from 0 to 6. For example, 3 marbles on the left looks like:

6!/[3!*(6-3)!] = 6!/[3!*3!] = 20

Applying this find the combinations for 0 through 6 marbles on the left yields this graph:

 

Assuming that the marble arrangements are random, any single combination is just as likely as the other ones, so set ups become more or less likely based on the number of ways to achieve them. For example, there is only one combo to have all marbles on the left side, so it has a relative probability of 1. To have 2 marbles on the left, however, has a probability 60 times greater since there are 60 combos that have 2 marbles on the left. This idea is used in physics a lot, but I don’t have the word limit to talk about statistical mechanics.

Counting is fun, and here was just a taste of it. There’s a lot more to combinatorics that melts the mind, but this intro holds a lot of important information.