$(2n + 1) (2n + 3) (2n + 5) ……. (4n – 1)$ is equal to :

Given $(1 – 2x + 5x^2 – 10x^3) (1 + x)^n = 1 + a_1x + a_2x^2 + ….$ and that $a_1^2\,= 2a_2$ then the value of $n$ is

If $a_r$ is the coefficient of $x^{10-r}$ in the Binomial expansion of $(1+x)^{10}$, then $\sum \limits_{r=1}^{10} r^3\left(\frac{a_r}{a_{r-1}}\right)^2$ 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………………

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 :