The term independent of $x$ in the expansion of ${\left( {{x^2} - \frac{{3\sqrt 3 }}{{{x^3}}}} \right)^{10}}$ is

Coefficient of $t^{20}$ in the expansion of $(1 + t^2)^{10}(1 + t^{10})(1 + t^{20})$ is

If the coefficients of $x^4, x^5$ and $x^6$ in the expansion of $(1+x)^n$ are in the arithmetic progression, then the maximum value of $n$ is :

The coefficient of ${x^{100}}$ in the expansion of $\sum\limits_{j = 0}^{200} {{{(1 + x)}^j}} $ is

If the coefficient of $x ^{15}$ in the expansion of $\left(a x^3+\frac{1}{b x^{\frac{1}{3}}}\right)^{15}$ is equal to the coefficient of $x^{-15}$ in the expansion of $\left(a x^{\frac{1}{3}}-\frac{1}{b x^3}\right)^{15}$, where $a$ and $b$ are positive real numbers, then for each such ordered pair $(a, b) :$

Find the cocfficient of $a^{5} b^{7}$ in $(a-2 b)^{12}$