The term independent of $x$ in the expansion of ${\left( {{x^2} - \frac{1}{x}} \right)^9}$ is

The coefficient of ${x^{53}}$ in the following expansion $\sum\limits_{m = 0}^{100} {{\,^{100}}{C_m}{{(x - 3)}^{100 - m}}} {.2^m}$is

Let the coefficients of $x ^{-1}$ and $x ^{-3}$ in the expansion of $\left(2 x^{\frac{1}{5}}-\frac{1}{x^{\frac{1}{5}}}\right)^{15}, x>0$, be $m$ and $n$ respectively. If $r$ is a positive integer such $m n^{2}={ }^{15} C _{ r } .2^{ r }$, then the value of $r$ is equal to

For the natural numbers $m, n$, if $(1-y)^{m}(1+y)^{n}=1+a_{1} y+a_{2} y^{2}+\ldots .+a_{m+n} y^{m+n}$ and $a_{1}=a_{2}$ $=10$, then the value of $(m+n)$ is equal to:

Find the term independent of $x$ in the expansion of $\left(\sqrt[3]{x}+\frac{1}{2 \sqrt[3]{x}}\right)^{18}, x\,>\,0$

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