If \( C_{r} \) stands for \( { }^{n} C_{r} \), then the sum of the series \[ \frac{2\left(\frac{...

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If \( C_{r} \) stands for \( { }^{n} C_{r} \), then the sum of the series
\[
\frac{2\left(\frac{n}{2}\right) !\left(\frac{n}{2}\right) !}{n !}\left[C_{0}^{2}-2 C_{1}^{2}+3 C_{2}^{2}-\ldots+(-1)^{n}(n+1) C_{n}^{2}\right]
\]
where
\( n \) is an even positive integer is
(a) \( (-1)^{n / 2}(n+2) \)
(b) \( (-1)^{n}(n+1) \)
(c) \( (-1)^{n / 2}(n+1) \)
(d) None of these
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