Grouping setup questions | Video lesson

- [Instructor] Now we're going to work through the questions for our grouping setup. Make sure that you watched the video in which I walked through the initial setup and deductions, before you continue with this video. And at any time, feel free to pause the video, if you wanna try a question before I explain it. This first question is an orientation question, and you usually get one of these per setup. It reads, the student could take which one of the following groups of courses during the summer school session? In other words, we're asked to determine which choice is an acceptable list of courses that the student takes. That means that four of the choices, the wrong ones, will break at least one of the rules, and the answer won't break any of the rules. The great thing about these questions, is that you usually don't need any deductions to get a quick and accurate answer. We'll show you an approach that you can use in order to save as much time as possible. It's usually a good idea to start with the rules, instead of the choices. If we look at each rule and eliminate the choices that violate that rule, then no matter what the answer ends up being, we will have gone through all of the rules only once. Let's see how to do that. Our first rule tells us that if history is in, then statistics and music can't be with history. If we look at the choices, which ones can we eliminate? Well a is out, because history and statistics are both listed in a. We can also eliminate b, since history and music are both listed in b. We never have to consider those two choices again, and we've only gone through one rule. The second rule is that if music is in, then physics and theater are both out. Looks like we can eliminate e, which shows music and theater both being in. And the last rule is that if writing is in, then physics and statistics are out. Here, we can cross off d, because we see in that choice that writing and physics are both in. That means that c is the only choice that's left, so it is our answer. To summarize what we just did for this orientation question, we started with the rules, and eliminated choices that violate each rule. If we had started with the choices, we would've had to go through each rule three times, in order to get to the answer. Instead, by starting with the rules, we only had to go through the rules once. The wonderful thing about this kind of question, is that they're very common, and they're a good one to get out of the way first, since they usually don't require any deductions or drawing. Here's a maximum question. We're asked to figure out what's the highest number of courses the student can take, without breaking any rules. These can often be pretty challenging, but let's work through this question one piece at a time. We know from the introduction that the student has to take at least three courses, and now we have to figure out what the highest number of possible courses could be. It's really good that we thought about eliminating elements when we were making deductions. Because, if you remember which course we said had the highest impact, it was music. We recognize that if music is chosen, then a whole bunch of courses can't be chosen. So since our goal is to maximize the courses, we wanna make sure not to select music. So let's put music in the out column. And that takes care of choice a. Let's think about the opposite extreme. Which course can we certainly choose without worrying that it will affect our maximum? Linguistics, so let's place linguistics in, since linguistics doesn't have any relationship with any of the courses, so we can safely choose it any time we want. Okay, which of the remaining five courses is almost as free as linguistics? In other words, which course only affects one other course? That would be theater, so if we're wanting to maximize our number of courses, theater is going to be a safer bet than the other classes that knock out two courses each. Okay, let's be strategic here. We're always taking inventory of who's left, who's unaccounted for, right? And it looks like history, statistics, writing and physics, are unaccounted for, as of this moment. Well we definitely can't take all of them, because every single one of these courses knocks at least one other course out. But notice that these remaining elements, are all involved in rules together. We can pair them this way. We can say either history or statistics is in, since they can't both be in, and then either writing or physics is in, since those two can't both be in. There's no way that we can get more than two courses out of this list before, so we have our answer. The maximum number of courses the student can take is four, answer d. To recap, you will always have enough information in the rules, and in the questions to find your answer. Many times you'll have to be strategic though. In this case, we used our knowledge of eliminating elements, and freer elements, in order to find our maximum. If the student takes neither physics nor writing, then it could be true that the student also takes neither x, nor y. We're given a new condition to consider here. So, with a new and temporary condition to consider, let's re-draw our initial diagram, and establish that physics and writing are both out. It's common for students to feel stuck at this point in the question, because we don't know what happens when physics is out, or when writing is out. We only have rules that tell us what happens when the class is in. So we have to look beyond the rules, and once again consider the impact of the numbers. If physics and writing are both out, and each of the choices lists two more classes that the student wouldn't take, that would mean that four classes are out, and three are in, in this scenario. So that helps a ton. We can test each choice pretty quickly by writing down which courses are the ones that are selected. Then, because we're looking for a could be true, we'll expect that four of the choices will break the rules, and one of them will not. In choice a, the courses that are out are history, linguistics, physics and writing. And that means that music, statistics, and theater would be the courses that are in. Well that doesn't work, because music and theater cannot be taken together, according to rule number two. Let's eliminate this and move on. Choice b, the student would not be taking history, music, physics, and writing. So that leaves linguistics, statistics, and theater to be in. This works, so we can stop here on testing, select the answer, and move on. To go through the remaining wrong choices quickly, choice c leaves us with linguistics, music, and theater, and that breaks a rule because music and theater can't be together. Choice d leaves us with history, statistics, and theater, and that doesn't work because rule number one tells us that history and statistics can't be together. And choice e leaves us with history, linguistics, and music, and that doesn't work, because rule one again, tells us that history and music can't be together. B is the answer here, and to summarize, we couldn't have gotten this point so easily without staying vigilant about the number limitations. The answer was not obvious from the list of rules alone. Instead, the answer came from the numbers, as well as the rules. In this question, we're given new information to consider, because the question asks, if the student takes music, then which one of the following must the student also take? Because we have a new condition, we'll re-draw our initial diagram, and establish the condition by putting music in the in column. Now it's time to make deductions. We know from the rules that we have a lot of courses that can't be taken alongside music. And those courses are history, physics, and theater. Well, we've figured out three courses that can't be chosen, but what about the courses that are chosen? We're gonna have to dig a little deeper for more deductions, and to do that, let's remember to consider the numbers of the situation. So far, we've established four courses concretely. There are seven total courses, and we know that at least two of those seven need to be chosen in order to meet the minimum of three courses. Let's list the courses that aren't accounted for yet. We haven't yet placed linguistics, statistics, or writing. Excellent, do we know anything about these courses based on the rules? Well, linguistics isn't in any of the rules, we know that. Statistics, we know, can't be selected alongside history, or writing, aha! Statistics and writing can't both be selected because of the last rule. That means that linguistics is definitely in, and then the third course selected is either statistics or writing. And then the other, of statistics and writing, is out. So the question asks us, which course the student must also take. The answer is e, the student must take linguistics. All of the other choices list courses that either must be out, or could be out.