Which one of the following pairs CANNOT be the second and third students to perform, respectively?

Skylar on December 8, 2019

@Sabrinamc, maybe I can help explain it!

First, let's look at the game setup:

We are given 5 students to order: G, H, R, S, T.
Each will perform one at a time and only once, so our diagram looks like: __ __ __ __ __
Now let's breakdown the rules:
#1: H - R. From this, we can deduce that H cannot be last and R cannot be first,
#2: (G - T) -> (R - T and S - T). The contrapositive of this is: (T - R or T - S) -> (T - G).
#3: Either (H - S and H - T) OR (T - H and S - H).

Now, let's look move onto the questions.

QUESTION (1):
G __ __ __ __
Since Gloria is first, we know that G will be before T. According to Rule #2, this also means that R and S have to be before T. According to Rule #1, H has to be before R. Therefore, T has to go in the fifth spot because all of the other variables must come before it. This leaves us with:
G __ __ __ T
This only Gives us three possible ways to fill in spots 2-4, since H has to come before R.
GSHRT is the first possibility, but this is incorrect because, according to rule #3, H either needs to be before both S and T or after both S and T. It cannot be in the middle. This leaves us with only two valid options:
GHRST
GHSRT
Therefore, (D) is the only option that provides a possibility that is shown in one of these two scenarios.

QUESTION (2):
- Immediately, we can eliminate (C) because we know that R cannot be first since, according to Rule #1, it has to come after H.
- This same rule allows us to eliminate (D) because if S is in spot 1 and R is in spot 2, H is not before R.
- We can also eliminate (A) because if G is in spot 1 and T is in spot 2, G is before T but there is no way for R and S to also precede T. This violates Rule #2.
- Lastly, we can eliminate (E) because Rule #3 tells us that we either need both T and S before H, or both T and S after H. Therefore, we cannot have T in spot 1 and H in spot 2 because S would have to go after H.
- So, we are left with (B). HGRST is an example of this.

QUESTION (3):
We are looking for the answer that is impossible to make without violating the rules. A through D have valid examples, as listed below:
(A) HRSGT
(B) TGSHR
(C) HGSRT
(D) HRTSG
(E) is the correct answer because it would force G to be before T, which according to Rule #2, would cause S and R to both be before T as well. This is impossible since the answer choice places T in the fifth spot.

QUESTION (4):
Let's plug each answer choice into the respective spots and see if we can make a valid scenario.
(A) __ G H __ __
In this case, R must be in either spot 4 or 5 because Rule #1 states it must come after H. However, T and S must also fill the two spots after H because Rule #3 states they must both either come before or after H and there is only one open spot before H. So, neither R, T, nor S can fill spot 1. Therefore, this scenario is impossible and (A) is correct.

QUESTION (5):
S __ __ __ __
Let's create a valid scenario and see what we can eliminate: STGHR. This meets all the rules and eliminates (B) and (C).
Let's make another scenario: STHRG. This meets all the rules and eliminates (A) and (D).
Therefore, we are left with (E),

QUESTION (6):
__ __ __ ___ S
Since S is in the last spot and therefore cannot go before T, we need to make sure that G is not before T so that we do not violate Rule #2. We also know that H cannot be in the fourth spot because Rule #3 says H either needs to be before or after both S and T.
With these restrictions in mind, let's make a hypothetical and see what we can eliminate: HRTGS. This eliminates (A), (C), and (E).
Now, let's alter our hypothetical to make: HTGRS. This eliminates (D).
Therefore, we are left with (B).

Does this all make sense? Please reach out with any other questions and best of luck with your studies!