+0

very challenging @Cphill @Melody

0
46
1

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!

Feb 27, 2021