تغییرات و ترکیب

Permutations and Combinations

So now we’re gonna talk about the issue of permutations versus combinations. And you may be wondering, why haven’t we mentioned permutations yet? It would seem that this would be an important topic in counting. We’ve gone through all these counting lessons, and we haven’t even touched the idea of permutations. Well the funny thing is, we haven’t mentioned the word, but the idea is already covered.

Any counting problem that you would encounter that involves permutations like ideas, you could solve it already with the fundamental counting principal. And in fact, using the Fundamental Counting Principle’s almost always a more, a much more efficient way of handling a permutation problem, much better than using the formula.

So the funny thing about math on this test, people get formula-happy, they think, “Boy, is I know a formula then I really understand it”. And we really try to discourage that point of view. Memorizing a formula is not the same as mathematical understanding. Understanding a principle like the fundamental counting principle, that is mathematical understanding.

Now there is one formula, I will recommend related to permutations, and this is something that you have already seen in the Fundamental Counting Principle lesson. But if you have ‘n’ items and you want to find the orders of all ‘n’ of them, well that would be the ‘n’ factorial. So, for example, suppose you are putting five books on a shelf and you wanna know how many orders can we put these five books?

Well, the way you would find this, the shortcut rule is that would just be five factorial. And of course five factorial, we’ve run into this before, five factorial means five times four times three times two times one. If you multiply that out, that’s 120. So that would be one shortcut formula you can use.

Now, for a more general case, if we have a pool of ‘n’ items, and we select K and we’re going to put those K items selected from the pool of ‘n’ in different orders, there is flooding around that out there, a permutation formula, a more general permutation formula. If you wanted to, you could go to Google, you could search for it, you could find it.

We’re going to discourage you from using that formula. And the reason we’re going to to using it is because if you use the Fundamental Counting Principle, you’re actually gonna have a more efficient solution and you’re much less likely to make mistakes. You’re actually gonna be on average, much better off not even knowing that permutation formula.

So everything I’ve been saying here’s a little bit abstract. Let’s talk about a particular problem. How would you solve a particular problem? Here’s a classic permutation problem. Now keep in mind, permutation problems, they’re not very common. You may see one, maybe two on the test.

It may not even show up on the test at all. It is a relatively uncommon scenario, but it could arise. You could, you could get a problem just like this. So lets look at this problem. From a group of ten committee members, three will be randomly selected: one as chair, one as treasurer, one as secretary.

These three will form the board of the committee. How many different possible boards can be formed? Well it’s clear that order matters here. In other words, picking one person as the chair and then another person as the treasurer. Say A is the chair, B is the treasurer.

That’s going to be different from picking B as the chair, and A as the treasure. So if we swap the order of people around in these particular offices, we’ll get a different board. So order matters here. Now, many people would say well the way to do this is to use the permutation formula. Again, we’re saying use of that permutation formula, we feel is a longer way to do the problem.

That actually makes he problem harder and the Fundamental Counting Principle is actually a much more sleek and efficient method to use to solve this problem. So what do I mean by this? Well think about the Fundamental Counting Principle. We have three slots; chair, treasurer, secretary.

For that first slot, the chair person, there are ten people we could pick. We’re picking randomly. Any of the ten could be the chair person. Once we pick that chair person, well then that person’s out but we have nine people left and so any of those nine people could be the treasurer. Once we pick those two, then there are two people out, they’ve already been assigned, there are eight unassigned people and any of those eight could be the secretary.

And if we just multiply those, that gives us the number of boards. So nine times eight is 72 times ten is 720. There are 720 possible boards. So this a very sleek and efficient way to solve this problem using the Fundamental Counting Principle, which we have already covered in many videos already.

This is much more efficient than using a permutation formula. And that’s why we recommend against memorizing that general permutation formula. Now, let’s back up a bit and look at the bigger picture here. How do you know when you’re looking at a problem, do I use permutation, do I use combination, do I use the Fundamental Counting Principle?

And I will say, this is the hardest part about accounting problem. Knowing how to set it up, knowing which approach to take. Well, first of all, just keep in mind the big rule here, the big rule is the Fundamental Counting Principle. Every counting problem on the planet can be solved with the fundamental counting principle.

That’s why it’s called Fundamental, because it applies to every single counting problem. So, something that you can always use, it’s more basic, more flexible and more widely applicable than either permutations or combinations. Permutations and combinations are very, very special cases. They’re, they’re handy shortcuts to have in individual cases but they only work in those individual cases.

They’re not widely applicable the way that the Fundamental Counting Principle is widely applicable,. That’s big idea number one. So if we are picking from a group and there are not repeats and order doesn’t matter. So in other words, if I pick ACB or ABC or CBA, all of those count as the same selection.

We’ll then that’s a combination. And we use the combination formulas which we discuss in the previous videos. If we’re picking with no repeats in different orders of the final selection are meaningfully different. And that’s what we mean by saying order matters. Then its a permutation and then we’re going to use the Fundamental Counting Principle which we’ve already covered in previous videos and we’re through talking about it in this lesson.

Finally, I’ll say, this issue of order matters versus order doesn’t matter. This is actually very tricky. Now, sometimes it’ll be very explicit. Sometimes the text of the problem will say the order doesn’t matter, or the order does matter. Sometimes they’ll make it very clear.

But sometimes it’s not particular clear and you need to interpret the problem to figure this out. And it turns out that there’s no really quick way, there’s no short way to say this, it depends very much, it varies from problem to problem. And part of what’s going on here, here we need to talk about the brain for this second.

The, the, the left brain, the very ana, analytical logical brain, that’s the side of the brain that’s always asking, What’s the right thing to do? And many people who are studying for the test, they just wanna know, just tell me the right thing to do. Just tell me the right thing to do. That’s the left brain question.

The right brain question. Is the question, what is the right way to look at the problem? And if you’re asking that question, what is the right way to look at the problem? Well, once you find the right way to look at the problem, what you have to do is perfectly clear. And this is something very important to think about when you’re thinking about counting problems.

When you read the solutions, don’t rush through to what formula’s did they use? What do you have to do? What do you have to do to solve this problem? That’s the wrong question to ask. The right question to ask with a counting problem is this right-brain question, what is the right way to look at the problem?

How did they look at the problem? What were the perceptual choices they made in dissecting this problem and setting up, deciding okay, we’re gonna count these first, then this, then this. What were those choices? And by studying those choices and learning, you develop the pattern-matching skills that will allow you to interpret counting problems more efficiently.