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Two numbers \( x \) and \( y \) are selected at random from \( \{1,2, \ldots, 5 n\} \) without replacement. If \( p_{n} \) denote the probability that \( \frac{1}{5}\left(x^{2}+y^{2}\right) \) is a natural number, then
(a) \( p_{n}=\frac{9 n-1}{5(5 n-1)} \)
(b) \( p_{n}=\frac{4 n}{5(5 n-1)} \)
(c) \( \lim _{n \rightarrow \infty} p_{n}=\frac{9}{25} \)
(d) \( \lim _{n \rightarrow \infty} p_{n}=\frac{4}{25} \)
W)
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