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Intersection of Sets

Let ω be a ...

Question

Let $ω$ be a complex cube root of unity with $ω$ $\neq$ 1. A fair die is thrown three times. If $r_{1}, r_{2}$ and $r_{3}$ are the numbers obtained on the die then the probability that $ω^{r_{1}} + ω^{r_{2}} + ω^{r_{3}} = 0$ is

\frac{1}{18}

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\frac{1}{9}

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\frac{2}{9}

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\frac{1}{36}

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Solution

The correct option is C $\frac{2}{9}$
$r_{1}, r_{2}, r_{3} \in {1, 2, 3, 4, 5, 6}$
$r_{1}, r_{2}, r_{3}$ are if the form $3 k, 3 k + 1, 3 k + 2$
Required probability $\frac{3! \times^{2} C_{1} \times^{2} C_{1} \times^{2} C_{1}}{6 \times 6 \times 6} = \frac{6 \times 8}{216} = \frac{2}{9}$

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