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Question

Two numbers $x$ & $y$ are chosen at random from the set ${1, 2, 3, 4, . . . .3 n}$ . Find the probability that $x^{n} - y^{n}$ is divisible by $3$ .

\frac{(5 n - 3)}{(9 n - 3)}

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\frac{(5 n - 4)}{(9 n - 3)}

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\frac{(5 n - 3)}{(9 n - 5)}

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\frac{(3 n - 5)}{(5 n - 3)}

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Solution

The correct option is A $\frac{(5 n - 3)}{(9 n - 3)}$
$x^{2} y^{2} = (x y) (x + y)$
$147 . . . . . . 3 n - 2$
$258 . . . . . . 3 n - 1$
$369 . . . . . . 3 n$
Required probability $= \frac{^{n} C_{2} +^{n} C_{1}^{n} C_{1} +^{n} C_{2} +^{n} C_{2}}{^{3 n} C_{2}}$
$= \frac{3 n (n - 1) + 2 n^{2}}{3 n (3 n - 1)} = \frac{5 n - 3}{9 n - 34}$

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