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

Let ${ }^{n} C_{r}$ denote the binomial coefficient of $x^{r}$ in the expansion of $(1+ x )^{ n }.$

If $\sum_{ k =0}^{10}\left(2^{2}+3 k \right){ }^{ n } C _{ k }=\alpha .3^{10}+\beta \cdot 2^{10}, \alpha, \beta \in R$ then $\alpha+\beta$ is equal to ……. .

$\sum\limits_{n = 0}^4 {{{\left( {1009 – 2n} \right)}^4}\left( \begin{gathered}
  4 \hfill \\
  n \hfill \\ 
\end{gathered}  \right)} {\left( { – 1} \right)^n}$   is

If $(1 -x + x^2)^n = a_0 + a_1x + a_2x^2 + ……. + a_{2n}x^{2n}$, then $a_0 + a_2 + a_4 +……..+ a_{2n}$ is equal to

If ${}^{21}{C_1} + 3.{}^{21}{C_3} + 5.{}^{21}{C_5} + ……19{}^{21}{C_{19}} + 21.{}^{21}{C_{21}} = k$ Then number of prime factors of $k$ is