Let S be the set of 10-tuples (a_0, a_1, \dots, a_9) where each entry is 0 or 1, so S contains 2^10 10-tuples. For each 10-tuple s = (a_0, a_1, \dots, a_9) in S, let p_s(x) be the polynomial of degree at most 9 such that p_s(n) = a_n for 0 \le n \le 9. For example, p(x) = p_{(0,1,0,0,1,0,1,0,0,0)}(x) is the polynomial of degree at most 9 such that p(0) = p(2) = p(3) = p(5) = p(7) = p(8) = p(9) = 0 and p(1) = p(4) = p(6) = 1. Find \sum_{s \in S} p_s(10).

Guest Mar 30, 2021