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Combinations

Combinations, this is a fun topic. So far we have been looking mostly at problems in which we were selecting ordered arrangements. Sometimes order doesn’t matter. For example, suppose from a group of 20 individuals, we are gonna choose a group of three for a team of some kind.

A small group, drawn from a large group, is a combination. And in combinations, we don’t care about the order of selection. All we care about is the final arrangement that we get. Suppose we have six employees of equal rank in a workgroup, and we’re gonna choose randomly three of them for a task. Let’s say the three chosen are B, E, and F.

All that matters is who has been selected. It doesn’t matter at all. Which one was picked first or second. It doesn’t matter we picked B first and then F or we picked E first then B. It does not matter, all that matters is the resultant group. In combinations only the final selected group matters; the order in which they were selected for that group doesn’t matter at all.

If we have a group of n different individuals, and we randomly select r of these individuals, then the number of combinations devoted, is denoted with this notation. We read this as n choose r. Okay, so have a pool of n and we’re choosing r. For example, 8 choose 3 would be the number of different 3-person combinations we could select from a pool of eight people.

Before we discuss how to calculate this, I will point out a few things. First of all, think about what 7 choose 1 means. It means we have a group of seven people and we’re gonna choose one of them at random. Well, how many ways can we do that? Well, there’s seven people.

So we could pick any of the seven. There’d be seven different ways to do it. So, it’s the number of ways I can choose one person from a pool of seven, so it has to equal seven. And in general, n choose one has to equal n. That’s important idea number one.

Now think about 10 choose 4. This is the number of different 4-person combinations we could select from a pool of qq10. We’ll talk about what 10 choose 4 equals in a minute. For now, notice that when we create a group of 4, we also automatically create a group of 6.

The 6 people left behind who weren’t chosen. Every group of 4 corresponds to a group of 6. So there must be the same number of them. In other words, 10 choose 4 must equal 10 choose 6. There’s a one to one correspondence between groups of 4 and groups of 6 when we’re choosing from a pool of 10.

And similarly, 10 choose 3 would have to equal 10 choose 7. 10 choose 2 would have to equal 10 choose 8. And so forth. In general, if we select a group of R. From a pool of n, we automatically create another group of n minus r folks who are left behind.

Starting from a pool of n, there must be one group of n minus r for each combination of r. Therefore, n choose r must equal n choose n minus r. That’s a very important formula and that shows a deep symmetry to these, these ideas of combination of these numbers that we get for combinations. Now we can discuss how to calculate 10 choose 4.

Suppose we actually need the number. Think about 10 choose 4. We have a pool of 10 people, 10 different items, and we wanna select for ow many different sets of 4 could we pick? Well, with the fundamental counting principle, we’d start with that, we’d choose 10, then 9 choices for the second, then 8 for, for the third choice, and 7 for the fourth choice.

So 10 times 9 times 8 times 7. But that would, would be correct if order mattered, but because the order of the 4 selected didn’t matter, we have to divide by 4 factorial. So 10 choose 4 is that product divided by 4 factorial. Multiply it out, we’ll do a little canceling. Cancel the 9 and the 3, we get 10 times 3 times 7.

So that would be 10 times 21 or 210. And so 10 choose 4 equals 210. That’s how we actually calculate it. Now notice we got another one for free, because as we noted earlier, 10 choose 4 has to equal 10 choose 6. So 10 choose 4 equals 210, and 10 choose 6 equals 210 also.

Here’s a practice problem. Pause the video and then we’ll talk about this. Okay, we have 12 different major rides. We can choose any of the 3 of them. So this is simply going to be, 12 choose 3.

So we just have to calculate 12 choose 3. So, first we’ll figure out we have 12 choices for the first one, 11 for the second, and 10 for the third. And then we divide by 3 factorial because the order doesn’t matter, so we’re limiting the repetitions due to the different orderings of the choices. And so 12 choose 3 would just be 12 times 11 times 10 divided by 6.

Well, 12 divided by 6, of course, we can cancel. We have 2 times 11 times 10, that’s 22 times 10 or 220. That’s how many sets of three rides we can do and that is the number equal to 12 choose 3, incidentally, would also be the number for 12 choose 9. Sometimes on the more advanced problems you may need to know the abstract combinations formula.

This is the abstract formula. And this formula is designed to have a massive amount of cancellation in it. But I will say, the formula is clunky. After a lot of cancellation, all that reduces to the much simpler fundamental counting principle approach that we were already using. So, this is one case in which, if you’re eager to memorize a formula, don’t.

This formula is not very useful. It would only be useful in a question that we’re asking very particularly about the formula. Yes, that could happen on a very advanced question, they could actually ask about the formula itself. Most practical combination questions, such as the ones we’ve done in the video, it’s much easier to use the fundamental counting principle.

And in fact, problems like this in some way are designed to punish the people who blindly rely on formulas rather than understanding more mathematical approaches to the problem. In summary, in a combination, we are concerned with an end-group of r individuals choose, chosen from a pool of, of n, and the order of selection does not matter.

n choose r is the number of ways to choose r individuals from a pool of n. n choose 1 always equals 1 for any value of n. n choose r always has to equal n choose n minus r. That’s a very important symm, symmetry formula. And we can calculate n choose r by starting with the fundamental counting principle and then dividing to eliminate repetitions.