The natural number m , for which the coefficient of $x$ in the binomial expansion of $\left(x^m+\frac{1}{x^2}\right)^{22}$ is 1540 , is $\_\_\_\_$ .

The total number of irrational terms in the binomial expansion of $\left(7^{1 / 5}-3^{1 / 10}\right)^{60}$ is :

If ${ }^{\mathrm{n}} \mathrm{C}_4,{ }^{\mathrm{n}} \mathrm{C}_5$ and ${ }^{\mathrm{n}} \mathrm{C}_6$ are in A.P., then n can be :

If $\{p\}$ denotes the fractional part of the number $p$, then $\left\{\frac{3^{200}}{8}\right\}$, is equal to:

If the constant term in the binomial expansion of $\left(\sqrt{x}-\frac{k}{x^2}\right)^{10}$ is 405 , then $|k|$ equals :

The coefficient of $t^4$ in the expansion of $\left(\frac{1-t^6}{1-t}\right)^3$ is

The coef ficient of $x^4$ in the expansion of $\left(1+x+x^2+x^3\right)^6$ in powers of $x$, is $\_\_\_\_$ ...

The sum of the real values of $x$ for which the middle term in the binomial expansion of $\left(\frac{x^3}{3}+\frac{3}{x}\right)^8$ equals 5670 is

If $a, b$ and $c$ are the greatest values of ${ }^{19} \mathrm{C}_{\mathrm{p}},{ }^{20} \mathrm{C}_{\mathrm{q}}$ and ${ }^{21} \mathrm{C}_{\mathrm{r}}$ respectively, then

The value of r for which ${ }^{20} \mathrm{C}_{\mathrm{r}}{ }^{20} \mathrm{C} 0+{ }^{20} \mathrm{C}_{\mathrm{r}-1}{ }^{20} \mathrm{C}_1+{ }^{20} \mathrm{C}_{\mathrm{r}-2}{ }^{20} \mathrm{C}_2+\ldots+{ }^{20} \mathrm{C}_0{ }^{20} \mathrm{Cr}$ is...