This is a rule substition question.  The best approach is to examine each answer choice and determine if it is "too strong" or "too weak."  "Too strong" means that the answer choice forces you to alter the diagrammatic layout of the game.  "Too weak" means that the answer choice doesn't account for how the game was originally laid out.  The correct answer produces a diagrammatic layout that is EXACTLY like the original layout.

"Haley and Peters cannot both be assigned to the East":  too weak--it leaves open the possibility of H and P being assigned to the West.

"If Haley is not assigned to the West, then Peters must be":  too weak--it doesn't actually replicate the original rule that bars H and P from being together under any circumstances.  This explanation also rules out the other two incorrect answer choices.

The new rule about no three of the specified volunteers being assigned together captures the original diagrammatic layout without any alteration.  So that is the correct answer.

This is a standard acceptability question, so the approach is to take each rule singly and eliminate rule violators.  The correct answer will be the one choice that is not stricken as a rule violator.

Keeping Manuel and Nina out of the production project requires the 2:1:5 distribution, since the 4:2:2 distribution couldn't maintain both blocks (Otis and Peter, Quinn and Reza).

We place one of Manuel and Nina on Sales, and the other as the only employee on Marketing. The only free employee who doesn't need to be paired with anyone at this point is Lisa, so she joins Sales. Quinn, Reza, Peter and Otis join Sam to round out Production.

If Quinn and Reza are both on the Marketing project, then it must be the 4:2:2 distribution. Since Sam is already on Production, Otis and Peter as a pair must be on Sales. This sets off the conditional that requires Lisa to be on the Production project, which then leaves Manuel and Nina to be together on the Sales project, which violates the rule.

This is a simple "grab-a-rule" questions - simply go through the answer choices looking for rule violations and eliminate accordingly. The conditional which states that if Peter is on Sales then Lisa is on Production eliminates one choice. The rule that Manuel and Nina cannot be together eliminates another. Quinn is not paired with Reza in one, which eliminates that choice. And Sam being assigned to Marketing instead of Production eliminates the final wrong choice.

If Peter is assigned to Sales, then we know Otis must be with him. This also sets off the conditional which states that Lisa must then be assigned to Production.

At this point it is important to understand the possible distributions of this game: if there must be EXACTLY twice as many people assigned to sales as marketing, then there are only two distributions: Sales: 2; Marketing: 1; Production: 5; OR Sales: 4; Marketing: 2; Production: 2.

At this point we have two in Sales (Peter and Otis) and two in Production (Sam and Lisa), so this could possibly work with either distribution. But if we try the second distribution and fill Sales with 4 employees, we quickly find that we can't split Manuel and Nina while still keeping Quinn and Reza together.

We therefore settle on the first distribution, with 2 in Sales, 1 in Marketing, and 5 in production. Peter and Otis alone fill sales. Either Manuel or Nina is alone in Marketing. Production is filled by Sam, Lisa, Quinn, Reza and the other one of Manuel/Nina.

While this question looks very intimidating, in reality it is actually a chance for you to solidify your understanding of the conditions of the game. No new information has been introduced, and all you have to do to determine the correct answer is "fact check" the information in the different responses. Start with the first response and apply each of the conditions from the pre-question text, one by one. As soon as you encounter an error, eliminate that answer and move on to the next one. For example, let's say that the first response went as follows:

"theater 1: romance, comedy, drama, and horror

theater 2: drama and thriller

theater 3: romance and horror"

We would start by referencing the first condition in the pre-question text:

"Exactly one theater will screen both the romance and the comedy." TRUE

So far, so good! In this case, theater 1 screens both the romance and the comedy. Let's go on to the second condition:

"Exactly two theaters will screen the drama." TRUE

Again, this response conforms to the conditions of the game. Theaters 1 and 2 both screen the drama. What about condition 3?

"If a theater screens the thriller, then that theater will screen neither the romance nor the horror." TRUE

3 for 3! Theater 2 will screen a thriller and a drama--no romance and no horror! This is the type of question where it pays to be thorough. It can be tempting to rush through in the hopes of finishing the game quickly, but as we'll see when we apply condition 4...

"If a theater does not screen the comedy, then that theater will screen the horror." FALSE

Theater 2 does not screen the comedy, and does not screen the horror. Eliminate this response and move on to the next one! 

The correct answer to this question is:

"theater 1: romance, comedy, and drama

theater 2: romance, drama, and horror

theater 3: comedy and thriller"

Theater 1 screens the romance and the comedy, theaters 1 and 2 screen the drama, theater 3 screens the thriller but not the romance or the horror, and theater 2 screens the horror but not the comedy. All conditions satisfied!

If you got this one right, good job! If not, try applying all the conditions again, and see if you can spot the place you went wrong. Questions like these are "gimmes", provided you take the time to carefully check all the information.

The temptation when we encounter a question like this is to draw out five versions of our game diagrams and meticulously fill in every possible combination of entities until we have determined beyond a shadow of a doubt which answer is correct. However, there is a much simpler way to solve this problem. We just have to think critically about the conditions in the pre-question text.

The first condition is actually satisfied by one of our responses: "romance and comedy". The other three conditions don't invalidate this response as a complete list, so we can eliminate it because it CAN be the complete listing of one of the theaters. That can get confusing, so it's important to remember that we are looking for the error here, and should eliminate any response that could work. 

The second condition will be too broad for our purposes, because we only have the information for one theater. You can lose a lot of time by trying to deduce the list of screenings in the other theaters to see if this condition is satisfied. Actually, your best bet in this situation is to move on and see if there is another condition that can help you eliminate answers more quickly.

The third condition is completely useless, as none of the theaters in our responses even screens the thriller. Even if we take the contrapositive of this condition, i.e., "If a theater screens EITHER the romance OR the horror, then it will NOT screen the thriller," it's just telling us what we can already observe in the responses. Move on.

The fourth condition is what this entire question hinges on. And if you are judicious in your assessment of the conditions, you will skip straight to the fourth condition and have your answer very quickly. "If a theater does not screen the comedy, then that theater will screen the horror." Remember that here "a theater" means "any theater," which means that in each theater being considered, you MUST have either the comedy of the horror. Just a cursory glance at the answers tells us that this condition is satisfied by every response choice save one: "romance and drama". It is impossible for this to be the complete list of screenings for any theater, as it includes neither the comedy nor the horror. This is our answer.

The question text for this particular problem is asking us for a list of screenings that could satisfy all the conditions, based on a complete list of screenings for theater 1. Unlike must be true questions, in which we eliminate all responses that are not always true, here we are simply looking for responses that are never true. The most efficient way to determine the correct solution is simply to go through the answers, applying each condition to them, and eliminating anything that either doesn't conform for theater 2, or has implications for theater 3 that would force it not to conform. Let's start with the correct solution:

The correct solution for this problem is: "romance, horror, and comedy".

Now let's combine this with the information from the question text, apply each condition, and see what results we get. 

In theater 1 (from the question text) we have comedy, drama, and horror. In the correct answer, we get romance, horror, and comedy for theater 2. Combined, we can see that so far our answers conform to condition 1 (exactly one theater will show both the romance and the comedy), and condition 4 (either horror or comedy or both). We can very easily plug in thriller, drama, and comedy for theater 3, and we will have satisfied the remaining conditions, along with the requirement that each film be screened at least once. For clarity's sake, here are all three theaters listed together:

theater 1 (question text): comedy, drama, and horror

theater 2: romance, horror, and comedy

theater 3: thriller, drama, and comedy

Very simple and straightforward solution to what sounds like a complex problem.

Now let's quickly break down why the other answers fail. Namely, none of the other responses satisfies the condition that the romance and the comedy be screened in the same theater. Because we haven't met condition 1 and we still haven't used the thriller, this forces us to use thriller, romance, and comedy in theater 3, which violates condition 3 (if thriller, no romance and no horror).

Try plugging in the information for all the incorrect answers and you will see that you hit this problem each time.

To determine the correct solution to this problem, we need to be able to think creatively about how the different conditions of the game are related to one another. We know from the question text that the horror film will be screened in more theaters than the romance, and from this we can immediately extract useful information. For one thing, neither the romance nor the horror can be screened in the same theater as the thriller. This effectively removes one theater from consideration, and allows us to quickly establish that the horror will be screened in two theaters and the romance in just one. Furthermore, when we consider that the romance and the comedy must be shown together in exactly one theater, we can already determine almost the exact distribution of films among theaters without even looking at the answers. It would look something like this:

theater 1: romance, horror, comedy/drama

theater 2: horror, comedy/drama

theater 3: thriller, comedy, (drama)

Because the only limitation on drama is that it appear exactly twice, for our purposes it can go in any two of the theaters. We also know that when we combine conditions 3 and 4, we are left with a theater 3 that screens at the minimum the thriller and the comedy, with the option to include the drama depending on how many theaters it has already been assigned to. 

Having taken very little time to map that out, we can now go to the answer section and see if any of the responses meet the criteria we've set. When we do so, we see that the correct solution is: horror and comedy. As you can see, this is exactly what we indicated in theater 2, only without including the optional drama screening.