If $C_{r}$ stands for $r^{n} C_{r}$ the sum of the series $\frac{2 (\frac{n}{2})! (\frac{n}{2})!}{n!} [C_{0}^{2} - 2 C_{1}^{2} + 3 C_{2}^{2} - \dots + (- 1)^{n} (n + 1) C_{n}^{2}]$
where n is an even positive integer, is equal to

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$(- 1)^{n / 2} (n + 1)$

$(- 1)^{n / 2} (n + 2)$

none of these

$(- 1)^{n} (n + 1)$

answer is A.

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Detailed Solution

Let $n = 2 m$ , and

$S = C_{0}^{2} - 2 C_{1}^{2} + 3 C_{2}^{2} - \dots +$

$(- 1)^{2 m} (2 m + 1) C_{2 m}^{2}$ (1)

Using $C_{r} = C_{n - r}$ , we rewrite (1) as

$\begin{aligned} S = (2 m + 1) C_{0}^{2} - (2 m) C_{1}^{2} + (2 m - 1) C_{2}^{2} - \dots + C_{2 m}^{2} \end{aligned}$ (2)

we get

$\begin{array}{l} 2 S = (2 m + 2) [C_{0}^{2} - C_{1}^{2} + C_{2}^{2} - \dots + C_{2 m}^{2}] \\ = (2 m + 2) (^{2 m} C_{m}) (- 1)^{m} \\ S = (m + 1) (^{2 m} C_{m}) (- 1)^{m} \\ = (- 1)^{n / 2} (\frac{n}{2} + 1) \frac{n!}{(\frac{n}{2})! (\frac{n}{2})!} \end{array}$