For $\mathrm{r}=0,1, \ldots, 10$, let $\mathrm{A}_{\mathrm{r}}, \mathrm{B}_{\mathrm{r}}$ and $\mathrm{C}_{\mathrm{r}}$ denote, respectively, the coefficient of $\mathrm{x}^{\mathrm{r}}$ in the expansions of $(1+\mathrm{x})^{10}$, $(1+\mathrm{x})^{20}$ and $(1+\mathrm{x})^{30}$. Then $\sum_{r=1}^{10} A_r\left(B_{10} B_r-C_{10} A_r\right)$ is equal to

If the fourth term in the Binomial expansion of ${\left( {\frac{2}{x} + {x^{{{\log }_e}x}}} \right)^6}(x > 0)$ is $20\times 8^7,$ then a value of $x$ is

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 coefficient of ${x^5}$ in the expansion of ${(x + 3)^6}$ is

The greatest value of the term independent of $x$ in the expansion of ${\left( {x\sin \theta  + \frac{{\cos \theta }}{x}} \right)^{10}}$ is

The coefficient of  $x^2$ in the expansion of the product $(2 -x^2)$. $((1 + 2x + 3x^2)^6 +(1 -4x^2)^6)$  is