If ${C_r}$ stands for $^n{C_r}$, the sum of the given series $\frac{{2(n/2)!(n/2)!}}{{n!}}[C_0^2 - 2C_1^2 + 3C_2^2 - ..... + {( - 1)^n}(n + 1)C_n^2]$, Where $n$ is an even positive integer, is

  • [IIT 1986]
  • A

    $0$

  • B

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

  • C

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

  • D

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

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