Textbook

Normally, a chapter gives you the basic information first and the more complex information later. In this case, it might seem like I’m doing it backwards, because this unit seems simpler than the previous one.

Bear with me, though.

- Those would found the last unit easy may find that this unit challenges certain assumptions that are helpful at first, but unhelpful later
- Those who found the last unit difficult may find that this helps it all click together.

The deeper you get into combinatorics, the more you will see the words “combination” and “permutation” used to describe various parts of problems. Let’s learn the basics, starting with **combinations**.

The 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 *different collections* 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.

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

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, as I mentioned in the last unit.

Example: “four combine two” ($_{4}C_{2}$) 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” ($_{4}P_{2}$), 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).

Continue to the next topic. There are no quizzes here.

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