Question

Consider the set {1,2,3,4,5,6,7,8,9,10}. For each subset, calculate the sum of the elements in the subset. How many distinct sums can we get?

Answer 1

all of the values from 1 to 10*11/2 = 55

Answer 2

There are 2^10 possible subsets, including Φ, the null set.
Not sure what you mean by sum of the elements, do you mean the sum of the cardinalities of the 1024 subsets, or do you mean the sum of the numbers of each subset?
In the case of the latter, the distinct sums would be from 1 to 55, as pointed out by Steve above. There is a certain difficulty with the null set, since there are no elements, we cannot obtain a sum.