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Sum minus product in Z_3^n is zero

Source: Canada Repêchage 2018/6

April 9, 2018
combinatorics

Problem Statement

Let n≥2n \geq 2 be a positive integer. Determine the number of nn-tuples (x1,x2,…,xn)(x_1, x_2, \ldots, x_n) such that xk∈{0,1,2}x_k \in \{0, 1, 2\} for 1≤k≤n1 \leq k \leq n and ∑k=1nxk−∏k=1nxk\sum_{k = 1}^n x_k - \prod_{k = 1}^n x_k is divisible by 33.
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