# Which one of the following pairs CANNOT be the fourth and fifth students to perform, respectively?

on January 27 at 10:33PM

Why is choice B wrong?

Victoria on January 27 at 11:30PM

Hi @Dina,

Happy to help!

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.