Daniel's symmetric sum

Algebra Level 5

Consider the set $S=\{1,2,3,\cdots,101\}$, and let $S_k$ be the sum of the product of every $k$ element subset of $S$ (also called the $k$th symmetric sum of $S$). Find the remainder when $\sum_{k=1}^{101} S_k$ is divided by $101$.

This problem is posed by Daniel C.

Details and assumptions

You may use the fact that 101 is prime.

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