Let $\alpha=\sum_{\mathrm{r}=0}^{\mathrm{n}}\left(4 \mathrm{r}^2+2 \mathrm{r}+1\right)^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}$ and $\beta=\left(\sum_{\mathrm{r}=0}^{\mathrm{n}} \frac{{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}}{\mathrm{r}+1}\right)+\frac{1}{\mathrm{n}+1}$. If $140<\frac{2 \alpha}{\beta}<281$ then the value of $n$ is...............

If ${ }^{20} \mathrm{C}_{\mathrm{r}}$ is the co-efficient of $\mathrm{x}^{\mathrm{r}}$ in the expansion of $(1+x)^{20}$, then the value of $\sum_{r=0}^{20} r^{2}\,\,{ }^{20} C_{r}$ is equal to :

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 :

If the sum of the coefficients in the expansion of ${(x - 2y + 3z)^n}$ is $128$ then the greatest coefficient in the expansion of ${(1 + x)^n}$ is

The sum of all the coefficients in the binomial expansion of ${({x^2} + x - 3)^{319}}$ is

If ${(1 + x - 2{x^2})^6} = 1 + {a_1}x + {a_2}{x^2} + .... + {a_{12}}{x^{12}}$, then the expression ${a_2} + {a_4} + {a_6} + .... + {a_{12}}$ has the value