For $S\subset\{1,2,…,117\}$ and $|S|=10$, let $A$ and $B$ be distinct subsets of $S$. $s_A$ is the sum of the elements in $A$ and $s_B$ is the sum of those in $B$. How can I prove that there must be at least one such pair for which $s_A=s_B$?
I see that the pigeonhole principle is going to come in handy here through some comparison between the least and greatest possible sums, but I'm rusty on the exact methods for this technique.
Best Answer
Suppose that $m$ is the smallest element of $S$. Let $t$ denote the sum of all elements of $S$. If $$t-m+1<2^{10}-1=1023\,,\text{ or equivalently }t-m\leq 1021\,,$$ then the claim follows, since there are $2^{10}-1$ nonempty subsets of $S$ whose element sums lie (inclusively) betweem $m$ and $t$. Now, prove that $t-m\leq 1021$ must hold.