Let $S$ be an arbitrary subset of $\{1, 2, …, 99\}$ with $|S|=10$. Prove that there are two different subsets $A$ and $B$ (don't have to be disjoint) of $S$ so that
$$\text{the sum of all the elements in $A$} = \text{the sum of all the elements in $B$}$$
Ex. $S=\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$, the sets $A = \{1, 2, 3, 4\}$ and $B = \{1,9\}$ satisfy the condition since $1 + 2 + 3 + 4 = 10 = 1 + 9$.
Edit: I know that the total subsets would be $2^{10} = 1024$. I originally thought that the largest sum value is $945$, but that wouldn't make sense because that implies that both A and B are the same set, which they can't be, so I don't know what number to compare it to.
Best Answer
However we choose our set the largest the sum of all of the numbers in the set can be is 945.
There are $2^{10} = 1024$ subsets of our set.
There are $1024$ pigeons and $945$ pigeon-holes. One pigeon-hole must contain more than one pigeon.