For which one of the following pairs of trucks must it be the case that at least one of the office buildings is serve...

Victoria on October 6, 2020

Hi @isabellaivanov,

Happy to help!

We know that there are six lunch trucks, each of which sells a different kind of food: F, H, I, P, S, or T.

Each truck serves one or more of exactly three buildings: X, Y, or Z.

F: _ _ _
H: _ _ _
I: _ _ _
P: _ _ _
S: _ _ _
T: _ _ _

Now let's go through our conditions.

Rule 1 - F, H, and exactly one other truck each serve Y.

This means that only one other truck can serve Y. All other trucks must serve either X or Z or both.

Rule 2 - F serves exactly two of the buildings.

This means that F serves either: (1) X and Y; or (2) Y and Z.

Rule 3 - I serves more buildings than S

This means S cannot serve three buildings and I cannot serve only one building.

Rule 4 - T does not serve Y

Therefore, either I, P, or S must serve Y.

Rule 5 - F does not serve any office building that P serves.

This means that P can only serve either X or Z (i.e. whichever one F does not serve).

Therefore, other I or S must serve Y.

Rule 6 - T serves two buildings that are also served by I. We know that T does not serve Y. Therefore, both I and T must serve both X and Z.

F: Y X/Z
H: Y _ _
I: X Z _
P: Z/X
S: _ _
T: X Z

If S serves Y, then I can only serve X and Z.

Now that we have our diagram, let's address the question stem. We are looking for the answer choice which must be true. It must be true that the two trucks in the answer choice serve at least one of the same office buildings.

Answer choice (A) does not have to be true because it is possible that H just serves Y and P just serves Z.

Answer choice (B) is incorrect because it is possible that H only serves Y and we know that T cannot serve Y.

Answer choice (D) does not have to be true because it is possible that I serves X and Z and S serves Y.

Answer choice (E) is incorrect because it is possible that S serves only Y and we know that T cannot serve Y.

Therefore, answer choice (C) should be our correct answer choice, but let's double check.

We know that I must serve X and Z. We also know that P must serve at least X or Z. Therefore, there must be at least one of these buildings that is served by both I and P.

Hope this helps! Please let us know if you have any further questions.