Rule 1 - at least one but no more than two of the CDs received each rating.
This means that it is possible that the six CDs only received three of the four possible ratings. It is also possible that they received all four possible ratings.
Rule 2 - H received exactly one more star than N did.
Therefore, H cannot receive one star and N cannot receive four stars. This means that we have three possibilities: H2N1, H3N2, and H4N3.
Rule 3 - either H or R received the same number of stars as I did.
H = I OR R = I
Rule 4 - at most one CD received more stars than Q did.
This means that it is possible that no CDs received more stars than Q. It is also possible that exactly one CD received more stars than Q did. This means Q must have received either 3 or 4 stars because if Q received 1 or 2 stars, then either every CD would have more stars than Q or we could only fit 5 CDs into our diagram, respectively.
The question stem asks us which of the answer choices must be true if R received the same rating as S.
If R received the same rating as S, then this rating is "full" because no more than two CDs can receive the same rating. Therefore, Rule 3 means that H must have received the same rating as I.
Let's map out our different possibilities based on Rule 2.
Option 1
1: N
2: H I
3: R S
4: Q
Our first option is H2N1. The question stem and our deduction made therefrom means that I must receive a rating of 2 stars. We know that R receives the same rating as S. Therefore, Q must receive a rating of 4 stars as Q1 or Q3 would violate Rule 4.
Option 2
1: R S
2: N
3: H I
4: Q
Our second option is H3N2. The question stem and our deduction made therefrom means that I must receive a rating of 3 stars. We know that R receives the same rating as S. Therefore, Q must receive a rating of 4 stars as Q1 or Q2 would violate Rule 4.
Option 3
1:
2:
3: N
4: H I
Our third option is H4N3. Notice that this option doesn't work as, even if Q receives a rating of 3 stars, the placement of H and I violates Rule 4.
Based on these three options, it seems like answer choice (D) - "Q received a rating of four stars" - must be true. However, let's run through the other answer choices to double check that we are right.
Answer choices (A) and (B) could be true, but they do not have to be true. Why? Because H and I must receive the same rating and we diagrammed two scenarios above where H and I can receive either two or three stars.
Answer choices (C) and (E) are incorrect for the same reason as above. They could be true, but they do not have to be true. Option 1 outlines a scenario where N receives one star and S receives three stars while still meeting all of the conditions.
Therefore, answer choice (D) must be true.
Hope this helps! Keep up the good work and please let us know if you have any further questions.
