The sum of the real values of $x$ for which the middle term in the binomial expansion of ${\left( {\frac{{{x^3}}}{3} + \frac{3}{x}} \right)^8}$ equals $5670$ is

The coefficient of middle term in the expansion of ${(1 + x)^{10}}$ is

Find the term independent of $x$ in the expansion of $\left(\frac{3}{2} x^{2}-\frac{1}{3 x}\right)^{6}$

Find the cocfficient of $x^{5}$ in $(x+3)^{8}$

The Coefficient of $x ^{-6}$, in the expansion of $\left(\frac{4 x}{5}+\frac{5}{2 x^2}\right)^9$, 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