We know that there are five students who will each perform exactly once at a concert: G, H, R, S, and T.
Let's start by going through the rules.
Rule 1 - H must perform earlier than R
H > R
Rule 2 - if G performs earlier than T, then R and S must also perform earlier than T
R G > T --> > T S
Contrapositive:
T > S or --> T > G T > R
Rule 3 - H must perform either earlier than both S and T or else later than both of them
> S H > T
OR
T > H S >
We are looking for the pair of students who cannot perform fourth and fifth, respectively.
What happens if H performs fourth and R performs fifth?
_ _ _ H R 1 2 3 4 5
This meets the first rule because H performs earlier than R.
To meet Rule 2, G cannot perform before T. Why? Because if G performs before T, then R must also perform before T. This is impossible because we know that R must perform last.
So, let's place G third with T and S performing first and second, respectively.
T S G H R 1 2 3 4 5
Notice that this meets all of the rules:
1) H performs before R
2) G does not perform earlier than T; therefore, this rule is irrelevant. If we look at the contrapositive, we can see that this is met. T performs earlier than S, R, and G. 3) H performs later than both S and T.
Therefore, it is entirely possible that H performs fourth and R performs fifth.
Answer choice (E) is our correct answer because it violates Rule 2. If T performs fourth and R performs fifth, then G must perform earlier than T. However, Rule 2 tells us that, if G performs earlier than T, then R must also perform earlier than T. This is impossible because we know that R must perform fifth.
Therefore, T and R cannot perform fourth and fifth, respectively.
Hope this helps! Please let us know if you have any further questions.
