Permutations & Combinations

I have now been blogging for 16 years, and my very first post (long gone) was on combinations and permutations. So, it’s fun to come back to the idea now. In 2004, my experience with the two concepts was limited to how textbooks often used the awkward “care about order” (permutations) or “don’t care about order” (combinations) language to introduce the ideas. So, that’s what I wrote about then. Now I want to talk about how the two concepts are related.

What They Are

When you count permutations, you count how many different ways you can sequentially arrange some things. When you count combinations, you count how many ways you can have some things. So, given 2 cards, there are 2 different ways you can sequentially arrange 2 cards, but given 2 cards, there’s just one way to have 2 cards.

Right off the bat, the language is weird, and it’s hard to see why combinations should ever be a thing (there’s always just 1 way to have a set of things). But combinations make better sense when you are not choosing from all the elements you are given.

So, for example, how many permutations and combinations can I make of 2 cards, chosen from a total of 3 cards?

Now having the two categories of permutation and combination makes a little more sense. There are 6 permutations of 2 cards chosen from 3 cards and there are 3 combinations of 2 cards chosen from 3 cards. That is, there are 6 different ways to sequentially arrange 2 cards chosen from 3 and just 3 different ways to have 2 cards chosen from 3. And you can see, by the way, that the combinations are a subset of the permutations.

In fact, let’s do an example with 4 cards to show the actual relationship between permutations and combinations. Here we’ll just use letters to save space. The permutations of JQKA if we choose 3 cards are: For permutations, we get:

JQK, JKQ, QJK, QKJ, KJQ, KQJ
JQA, JAQ, QJA, QAJ, AJQ, AQJ
QKA, QAK, KQA, KAQ, AQK, AKQ
KAJ, KJA, JKA, JAK, AJK, AKJ

That’s 24 permutations. For combinations, we get JQK, JQA, QKA, KAJ. That’s 4 combinations. What’s the relationship? We’ll come back on that next time.

Published by

Josh Fisher

Instructional designer, software development in K-12 mathematics education.