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Two numbers x and y are selected at random from the set ${1, 2, 3, \dots, 3 N}$ The probability that
$x^{2} - y^{2}$ is divisible by 3 is

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$(3 N - 1) / 3 N$

$(N - 1) / N$

$(5 N - 3) / (9 N - 3)$

$(N - 1) / 2 N$

answer is C.

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Detailed Solution

Use either both x and y are divisible by 3 or none of them is divisible by 3. Thus, required probability

$= \frac{(^{N} C_{2} +^{2 N} C_{2})}{^{3 N} C_{2}}$

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