Question 38
Let $S=\{1,2, \ldots, 2018\}$. For each subset $T$ of $S$, take the product of all elements of $T$, with $1$ being the product corresponding to the empty set. The sum of all such resulting products (with repetition) is $N$. Two elements $m$ and $n$ of $S$, with $m \lt n$, are removed. The sum of all products over all subsets of the resulting set is $\frac{N}{2018}$. What is $n$?