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When to Use Combinations
When to use combinations. This is a tricky issue. We use combinations when order doesn’t matter, okay? But what exactly does it mean for order to matter or not matter? Many students are confused on this issue of whether order matters in a particular problem.
Before we address the question of whether order matters, let’s be clear about why we’re asking it. If order matters in a problem, then most likely we will break the selection problem into stages, count the possibilities in each stage, and use the Fundamental Counting Principle. Just the straight Fundamental Counting Principle.
If order doesn’t matter, it’s a combination and we will use the combinations approach, that is, start with the Fundamental Counting Principle, but then we divide off the repetitions. So this whole business of whether to not, to divide off the repetitions, we divide off the repetitions if order doesn’t matter. So suppose we have a problem in which we have a pool of 20 committee members, and for the leadership team, we’re gonna pick three officers, a chairman, a treasurer, and a secretary.
Does order matter? Some students might be tempted to say no, because of course it doesn’t matter whether we pick the chairman first or the secretary first. But that’s not the right way to think about the question of whether order matters. One way to think about it is to be results oriented.
In other words, look at the group that results from the selection. If we can swap around the members of this selected group and this meaningfully changes the outcome, then order matters. For example, in this problem, suppose we got a leadership team of B as the chairperson, N as the treasurer, and F as the secretary. If we swap around the roles, make F the chairperson, B the treasurer, and N the secretary.
Well, that’s a different leadership team. In other words, different people in different roles, that’s a real difference. That’s a meaningful difference. So order actually does matter here. Because order matters, we would just use the ordinary Fundamental Counting Principle.
20 for the first choice, then there’d be 19 left for the second choice, and there’d be 18 left for the third choice. And so that would be the answer. We’re not gonna calculate that right now, that’s a larger number. And, in fact, the test might not even have you calculate that. The test might just leave it in, in a form of factors, like that.
But the point is, once we determine that order matters, then we break the problem into stages, calculate the number of options at each stage, and employ the Fundamental Counting Principle, and we’re done. That’s the answer. Suppose a chef is making a soup, and he has 20 vegetables from which to choose. He is instructed to chose any three vegetables, cut them into small pieces, and stir them into a large pot of soup.
How many different soups could he make? Well we’re assuming that the soup is, it’s determined purely by the choice of vegetables. Well here it doesn’t matter at all which vegetable is picked first or second or third because they will all be cut into small pieces and stirred together. So it’s absolutely irrelevant.
You’re just gonna have small pieces of one floating next to small pieces of the other. It doesn’t matter the order at all. They all wind up in the same pot. Because order doesn’t matter, it’s a combination. So this would be 20 choose 3, which would be 20 times 19 times 18 divided by radi, divided by 3 factorial.
Three factorial, of course, is six. We’ll do a little cancelling. Three times 19 is 57, so we have 20 times 57. Well, 20 times 50 is 1,000, and 20 times 7 is 140, so this would be 1,140, and that would be the total number of soups. These were two examples, but there is tremendous variety in counting problems.
Notice there is no clear-clut, clear-cut, black and white rule that you can always use to determine whether order matters. I’ve given you some guidelines, some ways of thinking about it, but there’s not a black and white rule system where you can always determine yes order matters, no order doesn’t matter. Ultimately determining whether order matters is what traditionally has been called a right-brain skill.
What does this mean? Well, according to that paradigm, the left-brain is analytical and logical. The left-brain loves clear rules and procedures. The left-brain’s question is this, what is the right thing to do? The left-brain loves nothing more than just an absolute fixed, clear recipe. GIve me an unambiguous recipe that I can follow.
That’s what the left-brain loves. The right-brain is said to be the world’s best pattern-matching device. The right-brain is non-linear and holistic. We dream with our right-brains, for example. The right-brain question are these. What is the right way to see the problem?
What is the right way to look at the problem? Or in other words, what is the right way to frame this situation? In counting problems, once you look at the problem the right way, what to do becomes easy to see. With counting problems, it’s very important to study the solutions looking for how to see.
That is, what was the right way to look at the problem? What perceptual choices did the author of the solution make at the very outset of the counting problem? So that can be one of the harder things to observe about a solution, especially if you’re rushing through just to, well, what should I do? What should I do?
Then you missed the most important part. The most important part is, well, how did they frame it at the very outset? How did they dissect it before they started doing anything? Here’s a practice problem. Pause the video and we’ll talk about this. Okay.
A classical concert will consist of two overtures in the first half and one symphony in the second half. A symphony and two overtures determine the program. The conductor can choose the program from 10 possible overtures and from 6 possible symphonies. How many programs are possible?
Well, let’s think about the overtures first. For the overtures, there are 10 choose 2 possibilities. Well, that is 10 on the first choice, 9 on the second choice, we divide by 2. That’s 5 times 9, which is 45, so there are 45 possible pairs of overtures that could be picked. And then there’s six possibilities for the symphony in the second half.
So we have the two overtures and the symphony, and it means multiply, so it’s going to be 6 times 45, we can use in doubling and halving. That’s 9, that’s 3 times 90. Three times 90, of course, would have to be 270. There are 270 possible programs. Here’s another practice problem.
Pause the video for a minute, and then we’ll talk about this. A librarian has a set of ten books, including four different books about Abraham Lincoln. The librarian wants to put the ten books on a shelf with the four Lincoln books next to each other, somewhere on the shelf.
How many different possible arrangements of the ten books are possible? This is a hard question. Since the four Lincoln books must be together, let’s start out by thinking of them as one big book. There are six books, and one big Lincoln book. How many different ways can we put these seven books in order on a shelf?
Well, of course, 7 factorial. That method of counting gives us a position for the Lincoln books among the other six books, but we still could put the four Lincoln books in any order. There are 4 factorial orders for the Lincoln books, so the total number of arrangements is 4 factorial times 7 factorial. That’s answer choice C.
Notice since in that, that we can’t simplify that any further. We have to leave that as 4 factorial times 7 factorial. In summary, if order madure, matters we break it into stages and just use the ordinary Fundamental Counting Principle. If order doesn’t matter, we use the combinations approach. One guideline is to consider the, the resulting arrangement and whether switching items around in this would res, would, in this result, would matter, whether it would meaningfully make some kind of different arrangement.
That’s one possible guideline to use. And, of course, you should read the solution sets and study how the instructor chooses to view the situation. In other words, what perceptual choices did the person writing the solution make in order to frame the question, in order to analyze it before they started doing any calculations?
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