If the OA is E, then the question seems to assume that the set DOES NOT consist of five distinct integers. Making that assumption:

S1:: The range is 3 or 7, so our set could be {1, 4, 4} or {3, 8, 10}; NOT SUFFICIENT.

S2:: We don't know the range; NOT SUFFICIENT.

Together, our set could be {1, 2, 4, 4, 4} or {3, 5, 7, 10, 10}; NOT SUFFICIENT.

The issue here is that (Sum of Set)/(# of terms) = Average.

S1 tells us the average = 3 or 7, S2 tells us the # of terms is 5. So we either have

Sum/5 = 3, so Sum = 15, or

Sum/5 = 7, so Sum = 35.

But if the Sum is 15, then our set MUST be 1+2+3+4+5. (This is the smallest possible sum of five distinct positive integers, so if the set has five different positive integers, it must be the set {1,2,3,4,5}.) But this set does NOT have a range of 3, so it cannot work. We could also deduce that five distinct positive integers forces a range of at least 4, so the range must be 7, etc.

Hence the sum of our set MUST be 35 ... which would lead to answer C.

S is a set of positive integers. The average of the terms in S is equal to the range of the terms in S. What is
the sum of all the integers in S?
(1) The range of S is a prime number that is less than 11 and is not a factor of 10.
(2) S is composed of 5 different integers.

There are many variables, but only 2 equations are given from the 2 conditions, making (E) our likely answer.
Looking at the conditions together,
sum/5=3,7, so sum=15, 35. This is insufficient, so the answer is (E).

For cases where we need 3 more equation, such as original conditions with "3 variables", or "4 variables and 1 equation", or "5 variables and 2 equations", we have 1 equation each in both 1) and 2). Therefore, there is 80% chance that E is the answer (especially about 90% of 2 by 2 questions where there are more than 3 variables), while C has 15% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since E is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, C or D.

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

S is a set of positive integers. The average of the terms in S is equal to the range of the terms in S. What is
the sum of all the integers in S?
(1) The range of S is a prime number that is less than 11 and is not a factor of 10.
(2) S is composed of 5 different integers.

If we modify the original condition, if we let elements' sum=S, elements' number=n, and range=r,
we need to know n and r, in conditions 1 and 2,
n=5, r=3,7, so this is insufficient and the answer becomes (E).

Once we modify the original condition and the question according to the variable approach method 1, we can solve approximately 30% of DS questions.