If ${(1 + x)^n} = {C_0} + {C_1}x + {C_2}{x^2} + .... + {C_n}{x^n}$, then ${C_0}{C_2} + {C_1}{C_3} + {C_2}{C_4} + {C_{n - 2}}{C_n}$ equals

Let $K$ be the sum of the coefficients of the odd powers of $x$ in the expansion of $(1+ x )^{99}$. Let a be the middle term in the expansion of $\left(2+\frac{1}{\sqrt{2}}\right)^{200}$. If $\frac{{ }^{200} C _{99} K }{ a }=\frac{2^{\ell} m }{ n }$, where $m$ and $n$ are odd numbers, then the ordered pair $(l, n )$ is equal to :

Let the coefficient of $x^{\mathrm{r}}$ in the expansion of $(\mathrm{x}+3)^{\mathrm{n}-1}+(\mathrm{x}+3)^{\mathrm{n}-2}(\mathrm{x}+2)+$ $(\mathrm{x}+3)^{\mathrm{n}-3}(\mathrm{x}+2)^2+\ldots \ldots+(\mathrm{x}+2)^{\mathrm{n}-1}$ be $\alpha_{\mathrm{r}}$. If $\sum_{\mathrm{r}=0}^{\mathrm{n}} \alpha_{\mathrm{r}}=\beta^{\mathrm{n}}-\gamma^{\mathrm{n}}, \beta, \gamma \in \mathrm{N}$, then the value of $\beta^2+\gamma^2$ equals..................

The coefficient of $x^8$ in the expansion of $(x-1) (x- 2) (x-3)...............(x-10)$ is :

The sum to $(n + 1)$ terms of the series $\frac{{{C_0}}}{2} - \frac{{{C_1}}}{3} + \frac{{{C_2}}}{4} - \frac{{{C_3}}}{5} + ...$ is 

The sum of the coefficients in the expansion of ${(1 + x - 3{x^2})^{3148}}$ is