very challenging @Cphill @Melody

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 \(\sum_{s \in S} p_s(10).\)
 

This took a long time to latex everything properly so i would appreciate quick and detailed solutions. Thanks very much!