Originally Posted by
Failure I suppose you have to show that there are more disjoint subsets of $\displaystyle S$ than there are possible different sums of the elements of those subsets. So the first question to answer is how many different pairs of disjoint subsets of $\displaystyle S$ there are.
Next, if you are lucky and there are a sufficiently large number of pairs of disjoint subsets of $\displaystyle S$, you might be able to use a very rough estimate of how many different sums there could possibly be (like, for example, $\displaystyle 10\cdot 99$).