jmc

The smallest $S$ is $1+2+\dots +90=91\cdot 45=4095$. The largest $S$ is $11+12+\dots +100=111\cdot 45=4995$. All numbers between $4095$ and $4995$ are possible values of S, so the number of possible values of S is $4995-4095+1=901$. Alternatively, for ease of calculation, let set $\mathcal{B}$ be a 10-element subset of $\{1,2,3,\dots ,100\}$, and let $T$ be the sum of the elements of $\mathcal{B}$. Note that the number of possible $S$ is the number of possible $T=5050-S$. The smallest possible $T$ is $1+2+\dots +10=55$ and the largest is $91+92+\dots +100=955$, so the number of possible values of T, and therefore S, is $955-55+1=\boxed{901}$.