First, I have to understand the question, and rephrase it if I can. Whenever I see any 2 of the words "sum", "average", or "number of terms", I have a feeling I'll be using the average formula here. So the question is asking me for the sum of all the integers in set S, which could also be asked as the average times the number of terms.
(1) Statement 1 tells me that the range is 2, 3, 5, or 7, and also that the range is not 1, 2, 5, or 10. Therefore, I know the range is 3 or 7. This is insufficient to tell me the sum of the terms, because their average could be 3 or it could be 7. I also don't know how many terms are in the set. Remove A and D from answer choices.
(2) Statement 2 tells me that the set has 5 different integers. Still, I don't know what those integers are or anything else about them, so I don't have sufficient information to answer the question. Remove B from answer choices.
When I try the two statements together, I know that the set has five different integers, so there's no way it could have a range of 3. The smallest the range could ever be for five different integers is 4! So now I know that I have a range of 7, which means I also have an average of 7. But, to find the sum of the set from the average, I'll also need the number of terms. And do you know that here?
Yes, you actually do. The word "set" means that nothing is repeated. (If the GMAT wanted to have repeating terms, they might call it something like a "list of integers".) So if S is a set with 5 different integers, it is a set of 5 terms. And while there are a few different sets that have 5 terms, a range of 7, and an average of 7, all those sets have the same sum. Remember, to find the sum of a set, just take the number of terms (5) and multiply by the average of those terms (7). So the sum here is definitely 35, and the answer is C.
