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 constant term, in binomial expansion of $\left(2 x^{r}+\frac{1}{x^{2}}\right)^{10}$ is $180,$ than $r$ is equal to $......$

The absolute difference of the coefficients of $x^{10}$ and $x^7$ in the expansion of $\left(2 x^2+\frac{1}{2 x}\right)^{11}$ is equal to

The number of integral terms in the expansion of ${({5^{1/2}} + {7^{1/6}})^{642}}$ is

If the non zero coefficient of $(2r + 4)th$ term is greater than non zero coefficient of $(r - 2)th$ term in expansion of $(1 + x)^{18}$, then number of possible integral values of $r$ is

The coefficient of $x ^7$ in $\left(1-x+2 x^3\right)^{10}$ is $........$.