Combinations and permutations | Combinatorics

The deeper you get into this field, the more you will see the words “combination” and “permutation” used to describe various parts of problems. We don’t need to get too far into this for the AMC 8, but let’s learn the basics.

The basic idea here is that you have some number of objects in a bag, and you plan to take a certain number of them out of the bag, and you want to count up the number of different ways in which you could do this. Another way of looking at this is that you are counting up the number of things you could end up with after you draw from the bag; that’s because each different way to do it gives you one particular thing at the end.

So for example, if you had three marbles in a bag (let’s say red, yellow, and blue), and you wanted to take two of them out, you might say that there are three ways to do that, because there are three different marble collections you could end up with in your hand: red and yellow, yellow and blue, red and blue. Three ways to do it; three possible collections in your hand at the end.

What we’ve just described is technically called a combination. And not just any combination, but a combination of two items drawn from a collection of three. We might say “a combination of two items taken from a set of three.” We might even shorthand this as “three combine two” or “three choose two,” both of which mean: “start with three things, and from that set, choose two of them.”

When we say these things, we aren’t actually talking about a particular collection of marbles we end up with. Instead, we’re talking about the number of different collections we could end up with.

So we’d say that “three choose two equals three.” Translation: if you start with three things in a bag, and you want to take two of them out, well, there are three ways you could do that. Which also means there are three different collections you might end up with.

We sometimes write $3 C 2 = 3$ and there are a few other ways to notate this. There is also a formula for this stuff, but that’s not important right at the moment. It’s the idea that matters most right now.

Okay, so I mentioned permutations also. What are those? Pretty much the same idea, except with permutations, you are not only keeping track of what you end up with, but also keeping track of the order in which you took them out of the bag. So if you’re doing the marble thing, but you’re tracking permutations, then you count “I draw a red marble and then a yellow marble” as different and separate from “I draw a yellow marble and then a red marble.”

So we’d say that “three permute two equals six,” meaning that if we have three marbles in a bag, and we want to pick two of them, and we care about what order we pick them in, then there are six ways to do this:

Red then yellow
Yellow then red
Red then blue
Blue then red
Yellow then blue
Blue then yellow

Again, there’s a formula, and again, we don’t care right at this moment.

The only thing we are going to point out is that if you have some situation where you are counting combinations, and you switch to counting permutations, your count will probably go way up. That’s because for any given combination, there are usually multiple permutations.

Example: “four combine two” counts the number of ways you could pick two marbles out of a bag of four of them. If you then consider counting “four permute two,” what you’ll see is that for every combination of two marbles (e.g. red and yellow) there are two different orderings --i.e. two different permutations-- of those two marbles, because you could get them in either of two ways (e.g. red then yellow, or yellow then red).

And, of course, if you are picking three marbles out of a bag, then there are lots of different orderings for each combination (six, in fact).