Daniel's symmetric sum

Algebra Level 5

Consider the set S={1,2,3,,101}S=\{1,2,3,\cdots,101\}, and let SkS_k be the sum of the product of every kk element subset of SS (also called the kkth symmetric sum of SS). Find the remainder when k=1101Sk\sum_{k=1}^{101} S_k is divided by 101101.

This problem is posed by Daniel C.

Details and assumptions

You may use the fact that 101 is prime.


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