# Prove or disprove whether there exists two subsets of $X$ such that sum of all the elements in each of these set is same

Let $$X\subset\{1,2,\ldots 100\}$$ be set with cardinality $$10$$. Then I need to prove or disprove whether there exist two subsets of $$X$$ such that the sum of all the elements in either of these sets is the same.

Total number of subsets of $$X$$ = $$2^{10} =1024$$

Also, the maximum sum of elements in a subset of $$X$$ can be $$100+99+\ldots 91 = 955$$

So by the pigeonhole principle, there must exist at least $$\big\lceil\frac{1024}{955}\big\rceil= 2$$ subsets that have the same sum

Is this correct?

Strictly speaking there are $$956$$ possible sums from $$0$$ to $$955$$. (You included the empty set in your 1024 sets.)

Other than that your proof is extremely good and clearly expressed.