## PrepTest 79, Game 3, Setup

### Transcript

An economics department is assigning six teaching assistants R, S, T, V, Y, and Z to three courses L, M, and P. Each assistant will be assigned to exactly one course and each course will have at least one assistant assigned to it. The assignment of assistants to courses is subject to the following conditions. Markets must have exactly two assistants assigned to it.

S and T must be assigned to the same course as each other. V and Y cannot be assigned to the same course as each other. Y and Z must be assigned to pricing if either of them is. All right, so we are dealing with a grouping game here. We have six teaching assistants and three courses, and each assistant is going to be assigned to at least one course.

And each course will, well, each assistant is gonna be assigned to exactly one course, and each course will have at least one assistant assigned to it. So there's no element of order here, so we're not dealing with any type of linear situation or a linear game. We're only concern with how the teaching assistant can be assigned to the courses, or which courses can be assigned to.

So let's look at our variables and make sure they are really clear on what those are. So we have our TAs which is gonna be R, S, T, V, Y, and Z. Then we have our courses L, M, and P. And then sort of our key information that each TA teaches one course and each course has at least one TA.

So when we were thinking about our diagram, we want to since we're trying to figure out which courses are being taught by which TAs it makes sense to sort of distribute the TAs into the courses. So, we will put our courses as the base of whatever diagram that we're able to come up with since is the most stable element and is not moving. So we'll end up with something that looks like this, of course, we have our variable set.

And then we have our three courses, L, M, or P. And then these are just sort of symbolizing the classes or the TAs that can be assigned there. So, of course, our rules may give us information about minimum or maximums. And we don't have this many variables, so some of these variables spaces would be blank.

But it's just giving a sort of a full scope of the possibilities for each of this courses. So now let's look at our rules. Rule #1, markets must have exactly two assistants assigned to it. So as we were just talking about, this is just a rough sort of estimate of the maximum capacity for each of these courses.

So we can go ahead and take one of those spots out of contention for marketing. So marketing can only have two, and it will have exactly two assistants assigned to it. Rule #2, S and T must be assigned to the same course as each other. So it doesn't really tell us much other than putting S and T into a blocked sequence here.

So wherever S teaches, T is gonna teach, vice versa. So those two need to go together wherever they go. Rule #3, V and Y cannot be assigned to the same course as each other, so this is a bit of the opposite of the previous rule, so we cannot have V and Y together. Rule #4, Y and Z must be both assigned to pricing if either one of them is. So what this rule is telling us is that if either Y or Z, either one of them are assigned to Pricing, then they both must be assigned to Pricing.

So if we have Y in Pricing or Z in Pricing, then we are gonna have Y and Z in Pricing. So this is only applicable when Y or Z is in P, so it's not saying that they need to occur together all the time, only when Y or P is in Z do we need to bring the other one along. So if one of those is in P both of them are in P, if one is in L we don't know where the other one is, so can't make any crazy assumptions there.

So now let's look at our master diagrams. So we have our variable set, we have two exactly going into course M. We have S and T occurring together, V and Y cannot occur together. And then we have our sort of conditional here, where if Y or Z is assigned to P, then both of them must be assigned to P. So, we have some restrictions on a lot of our rules and we have some sequencing or we have some conditional rules that can help us sort of determine given some information on what the possible arrangement of these courses may be.

So this should set you on your way to answering the questions in the section.