AMC 12 2017a Problem 25

The vertices $V$ of a centrally symmetric hexagon in the complex plane are given by $$V=\left\{ \sqrt{2}i,-\sqrt{2}i, \frac{1}{\sqrt{8}}(1+i),\frac{1}{\sqrt{8}}(-1+i),\frac{1}{\sqrt{8}}(1-i),\frac{1}{\sqrt{8}}(-1-i) \right\}.$$ For each $j$, $1\leq j\leq 12$, an element $z_j$ is chosen from $V$ at random, independently of the other choices. Let $P={\prod}_{j=1}^{12}z_j$ be the product of the $12$ numbers selected. What is the probability that $P=-1$?

$\textbf{(A) } \dfrac{5\cdot11}{3^{10}} \textbf{(B) } \dfrac{5^2\cdot11}{2\cdot3^{10}} \textbf{(C) } \dfrac{5\cdot11}{3^{9}} \textbf{(D) } \dfrac{5\cdot7\cdot11}{2\cdot3^{10}} \textbf{(E) } \dfrac{2^2\cdot5\cdot11}{3^{10}}$