2017 AMC 12A 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}} \qquad \textbf{(B) } \dfrac{5^2\cdot11}{2\cdot3^{10}} \qquad \textbf{(C) } \dfrac{5\cdot11}{3^{9}} \qquad \textbf{(D) } \dfrac{5\cdot7\cdot11}{2\cdot3^{10}} \qquad \textbf{(E) } \dfrac{2^2\cdot5\cdot11}{3^{10}}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.