jmc

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 $$\sum _{s\in S}{p}_s(10).$$