question_answer

If \[{{C}_{r}}\]stands for \[{}^{n}{{C}_{r}},\] then the sum of the series\[\frac{2\,\left( \frac{n}{2} \right)!\left( \frac{n}{2} \right)!}{2!}[C_{0}^{2}-2\,C_{1}^{2}+3\,C_{2}^{2}-.......+{{(-1)}^{n}}\,(n+1)\,C_{n}^{2}]\]. Where n is an even positive integer, is equal to:

A)  \[{{(-1)}^{n/2}}\,(n+1)\]                     

B)  \[{{(-1)}^{n}}\,(n+1)\]

C)  \[{{(-1)}^{n/2}}\,(n+2)\]         

D)  \[{{(-1)}^{n}}\,(n+2)\]

E)  None of these