If $\alpha$ and $\beta$ be the coefficients of $x^4$ and $x^2$ respectively in the expansion of $\left(x+\sqrt{x^2-1}\right)^6+\left(x-\sqrt{x^2-1}\right)^6$, then

If the term independent of $x$ in the expansion of $\left(\frac{3}{2} x^2-\frac{1}{3 x}\right)^9$ is $k$, then $18 k$ is equal to :

If the sum of the coefficients of all even powers of x in the product
$\left(1+x+x^2+\ldots \ldots+x^{2 n}\right)\left(1-x+x^2-x^3+\ldots .+x^{2 n}\right)$
Is 61, then...

If some three consecutive coefficients in the binomial expansion of $(x+1)^n$ in powers of $x$ are in the ratio $2: 15: 70$, then the average...

If $\left(\frac{3^6}{4^4}\right) k$ is the term, independent of x , in the binomial expansion of $\left(\frac{x}{4}-\frac{12}{x^2}\right)^{12}$, then k is equal to

If the coefficient of $a^7 b^8$ in the expansion of $(a+2 b+4 a b)^{10}$ is $K .2^{16}$, then $K$ is equal to $\_\_\_\_$ .

If $\sum_{\mathrm{r}=0}^{25}\left\{{ }^{50} \mathrm{C}_{\mathrm{r}} \cdot{ }^{50-\mathrm{r}} \mathrm{C}_{25-\mathrm{r}}\right\}=\mathrm{K}\left({ }^{50} \mathrm{C}_{25}\right)$, then K is equal to:

The positive value of $\lambda$ for which the co-efficient of $x^2$ in the expression $x^2\left(\sqrt{x}+\frac{\pi}{x^2}\right)^{10}$ is 720 , is

$\sum_{\mathrm{k}=0}^{20}{ }^{20} \mathrm{C}_{\mathrm{k}} \cdot{ }^{20} \mathrm{C}_{20-\mathrm{k}}$ is equal to :

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