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Let $α$ be a root of $x^{2} + x + 1 = 0$ and suppose that a fair die is thrown 3 times. If a, b, c are the numbers shown on the die, then the probability that $α^{a} + α^{b} + α^{c} = 0$ is

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$\frac{1}{72}$

$\frac{2}{36}$

$\frac{2}{9}$

$\frac{1}{27}$

answer is D.

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Detailed Solution

$α$ is root of $x^{2} + x + 1 = 0 \Rightarrow α = w$

a fair die thrown 3 times

$\begin{array}{rcl} n (S) & = & 6^{3} = 216 \\ E & = & (a, b, c) : w^{a} + w^{b} + w^{c} = 0 \end{array}$

$\begin{matrix} a & b & c \\ 3 & 1 & 2 \\ 3 & 4 & 5 \\ 6 & 1 & 2 \\ 6 & 4 & 5 \end{matrix} \begin{matrix} n (E) = 24 \times 2 \\ P (E) = \frac{24 \times 2}{216} \end{matrix} = \frac{8}{36} = \frac{2}{9}$

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