2018 AMC 10A Problem 18

Note that every number formed by this sum is either positive, negative, or zero. The number of positive numbers equals the number of negative numbers due to symmetry (flip the $1$ s to $-1$ s and $-1$ s to $1$ s). The only way for the sum to be $0$ is if all the coefficients are $0.$ The total number of numbers is $3^8 = 6561.$ Because each power of $3$ is larger than the sum of all previous powers of three, each combination of coefficients yields a different value. More explicitly, at the highest place where two combinations differ, the difference has magnitude at least $3^k,$ while all lower places together can cancel at most $2(1+3+\cdots+3^{k-1})=3^k-1.$ Therefore, there are $$\dfrac{6561 - 1}{2} + 1 = 3281$$ distinct nonnegative integers. Thus, D is the correct answer.