Mixed setup | Overview | Rules and deductions

- [Instructor] We're now going to work through a mixed setup together, so make sure that you've already worked through ordering setups and grouping setups before you tackle this one, since we're going to be combining the two tasks into one setup here. So let's start by examining the introductory passage to understand what we're being asked to do. The passage tells us a radio station is planning a program in which a total of six politicians, Fallon, Greer, Hernandez, Kim, Lewis and Munson, will be interviewed. The program will consist of exactly four segments. At least one of the politicians will be interviewed in each segment, and none will be interviewed in more than one segment. The following constraints apply. Before we look at the rules, let's understand what's going on and then build a rough sketch to reflect what we understand the situation to be. So first, our players are six politicians, so it's a pretty good idea to take inventory of them by writing a list of their initials. We have Fallon, Greer, Hernandez, Kim, Lewis, and Munson. Next, we can ask ourselves what we're doing with these politicians, and in this setup, we're grouping them into four segments. Since the order of the segments matters, given that this is one single program that's broken into four segments, that means that this is a mixed setup. If we had had four politicians and four segments, one right after another, then this would actually have been a classic ordering setup. But in this case, we have six politicians for four segments, so that means that some of them will have to be grouped together. Now, for a mixed setup that involves grouping and ordering like this, it can be really helpful to look at the ordering part first, and then try to figure out how the grouping aspect fits in. So, knowing that we have four segments, we can draw a blank for each segment and number them. All right, now let's think about the grouping. With six politicians and four segments, how many blanks do we still have left to place? Two. What does that mean for our possibilities? Well, either one segment gets three politicians and the rest of the segments get one politician each, or two segments get two politicians each, and then two segments get one politician each. So we're looking at three, one, one, and one, in some kind of order, or two, two, one, and one, and again, we wouldn't know the order. So now we know that the maximum number of politicians interviewed in any given segment has to be three, and we can mark that if we're worried that we'll forget. So for our diagram, if we were to find out that a segment had more than one politician, we could draw additional blanks underneath the corresponding segment number, and that gives us a really good starting framework in terms of the numbers, and that we can hope that the rules will give us information that allows us to deduce even more about the numbers. At this point we have a basic diagram, so let's go ahead and move on to the rules. Our first rule is an ordering rule, and it tells us that Hernandez must be interviewed in a segment that is earlier than any segment in which Fallon or Munson is interviewed. Well, what does this mean? It means that Hernandez is an earlier segment than Fallon, and that Hernandez is an earlier segment than Munson. So we can make a note of this so we don't forget, like this. It's important to note that we don't know how Munson and Fallon relate to each other. Munson and Fallon could be on the same day, or they could be on separate days, with either Munson earlier than Fallon, or Fallon earlier than Munson. Even with these uncertainties, we can start making deductions right away. So, think: Who can't be in the fourth segment based on this rule? Hernandez can't be in the fourth segment, since Hernandez has to be in an earlier segment than at least two people. Be careful though. Hernandez could be in the third segment if Fallon and Munson are together in the fourth segment. So this is all we know so far about where Hernandez can't go. What about Fallon and Munson? We know that they can't be in the first segment, again, because Hernandez has to be in an earlier segment than both of them, which would be impossible if either Fallon or Munson were first. Our second rule is a grouping rule, and it tells us that Kim and Munson must be interviewed in the same segment as each other. Normally we would note down a separate rule that Kim and Munson are a pair, but here we can actually save a bit of time by noticing that we just learned about Munson in the first rule. So we can actually combine rule number two with rule number one like this. And our group of politicians is getting bigger, which means we know more. In which segment can Kim not be interviewed? In segment one, because Kim has to be sometime later than Hernandez. We now know of three politicians who can't be in segment one. The third and final rule is that more of the politicians must be interviewed in the first segment than in the second segment. This is huge, it tells us a lot. We can deduce that the first segment must have at least two politicians, and maybe it even has three. And if the first segment had three politicians, then all of the other segments would have to have one politician each. So that means we know that the only segment that even could have three politicians is the first one, since the numbers combinations are either two, two, one, one, or three, one, one, one. We also know that if the first segment had two politicians and not three, then the second segment would still have only one politician so we can deduce that the second segment must have only one politician in any case, and we can mark this in any number of ways. I like closing off the column with double lines to remind myself that no more blanks can ever be placed there. You can choose whatever works best for you. Let's see what else we can deduce by revisiting some of the rules, now that we have some stronger number guidelines. We learned that Munson and Kim are together, and that they're sometime after Hernandez. Well, we just deduced that there can be only one politician interviewed in segment two. This tells us that the Munson and Kim pair, which we know can't be in the first segment, must be together at some point in the third or fourth segment. So, it's no longer a possibility that there are three politicians in the first segment. We know that there must be only two politicians in segment one, and either two politicians in segment three or in segment four. That means that we're limited to only two scenarios. So if you'd like, we can build them out in order to see everything more clearly. (virtual marker squeaks) In scenario one, we'll put Munson and Kim in segment three, and in scenario two, we'll put Munson and Kim in scenario four. These are the only two places that the Munson and Kim pair can go, which is why it's a good idea for us to build out these two scenarios. Now we can see a lot more. What else? Since we now know that no segment can include three politicians, we can look at our grouping here and see that Fallon and the Munson-Kim pair can't be together. Why is this important? Well, since they have to be on separate days, that means that there are at least two segments after Hernandez is interviewed, so now we know that Hernandez can't be interviewed on day three. To recap, we just set up a mixed task that involves ordering and grouping. We took the numbers guidelines that we were given in the last rule and made sure to get all of the information from it that we possibly could, and that allowed us to understand that only two scenarios involving Munson and Kim are possible. We took our time and looked at the implications of each rule making sure to included deductions about where elements can't go, not just where they can go. So we are in a really strong position to move to the questions with this initial diagram that we're gonna use for support.