In the expansion of ${\left( {x - \frac{3}{{{x^2}}}} \right)^9},$ the term independent of $x$ is

The positive value of $a$ so that the co-efficient of $x^5$ is equal to that of $x^{15}$ in the expansion of ${\left( {{x^2}\,\, + \,\,\frac{a}{{{x^3}}}} \right)^{10}}$ is

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 coefficients of the three successive terms in the binomial expansion of $(1 + x)^n$ are in the ratio $1 : 7 : 42,$ then the first of these terms in the expansion is

The sum of the coefficients of the first three terms in the expansion of $\left(x-\frac{3}{x^{2}}\right)^{m}, x \neq 0, m$ being a natural number, is $559 .$ Find the term of the expansion containing $x^{3}$

If $^n{C_{r - 2}} = 36$ , $^n{C_{r - 1}} = 84$   and    $^n{C_r} = 126$ , then value of $^n{C_{2r}}$ is