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How many degree $\leq$ $d$ mod($p$) polynomials are there such that $P(a_1),...,P(a_k) \neq 0$ for $k < d$ and $0 < a_1 <...< a_k < p$, all integers? I considered subtracting out elements from the the set of all polynomials of degree less than or equal to $d$, but this didn't get me very far.

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Hint: Apply the technique Principle of Inclusion and Exclusion.

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