Let \( S=\{1,2,3, \ldots \ldots n\} \) be a set and two numbers \( x \) and \( y \) are drawn wi...

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Let \( S=\{1,2,3, \ldots \ldots n\} \) be a set and two numbers \( x \) and \( y \) are drawn with replacement from \( S \). Let \( P_{2} \) be the probability that \( x^{2}-y^{2} \) is divisible by \( 2 \&[\cdot] \) denotes greatest integer less than or equal to \( x \).
The value of \( \mathrm{P} 2 \) is equal to
(a) \( \frac{1}{2} \)
(b) \( \left(\frac{\left[\frac{n}{2}\right]}{n}\right)^{2} \)
(c) \( 1-2 . \frac{\left[\frac{n}{2}\right]}{n}+2\left(\frac{\left[\frac{n}{2}\right]}{n}\right)^{2} \)
(d) \( 1-\frac{2-\left[\frac{n}{2}\right]}{n}-2\left(\frac{\left[\frac{n}{2}\right]}{n}\right)^{2} \)
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