MathDB

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

<span class='latex-bold'>(A) </span> \dfrac{5\cdot11}{3^{10}} \qquad <span class='latex-bold'>(B) </span> \dfrac{5^2\cdot11}{2\cdot3^{10}} \qquad <span class='latex-bold'>(C) </span> \dfrac{5\cdot11}{3^{9}} \qquad <span class='latex-bold'>(D) </span> \dfrac{5\cdot7\cdot11}{2\cdot3^{10}} \qquad <span class='latex-bold'>(E) </span> \dfrac{2^2\cdot5\cdot11}{3^{10}}

Problem Statement