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...

Let $\binom{n}{k}$ denotes ${ }^n C_k$ and $...

$3 \times 7^{22}+2 \times 10^{22}-44$ when divided by 18 leaves the remainder $\_\_\_\_$

The sum of the series $2 .{ }^{20} \mathrm{C}_0+5 .{ }^{20} \mathrm{C}_1+8 .{ }^{20} \mathrm{C}_2+ 11 .{ }^{20} \mathrm{C}_3+\ldots+62 .{ }^{20} \mathrm{C}_{20}$ is equal to

The sum of the co-efficients of all even degree terms in x in the expansion of $\left(x+\sqrt{x^3-1}\right)^6+\left(x-\sqrt{x^3-1}\right)^6,(x>1)$ is equal to :

If the fractional part of the number $\frac{2^{403}}{15}$ is $\frac{k}{15}$, then $k$ is equal to

If the sum of the coefficients in the expansion of $(x+y)^n$ is 4096 , then the greatest coefficient in the expansion is $\_\_\_\_$ .

The term independent of ' $x$ ' in the expansion of $\left(\frac{x+1}{x^{2 / 3}-x^{1 / 3}+1}-\frac{x-1}{x-x^{1 / 2}}\right)^{10}$, where $x \neq 0,1$ is equal to

The ratio of the coefficient of the middle term in the expansion of $(1+x)^{20}$ and the sum of the coefficients of two middle terms in...

If $b$ is very small as compared to the value of $a$, so that the cube and other higher powers of $\frac{b}{a}$ can be neglected...