The coefficient of ${x^{32}}$ in the expansion of ${\left( {{x^4} - \frac{1}{{{x^3}}}} \right)^{15}}$ is

Let the coefficients of third, fourth and fifth terms in the expansion of $\left(x+\frac{a}{x^{2}}\right)^{n}, x \neq 0,$ be in the ratio $12: 8: 3 .$ Then the term independent of $x$ in the expansion, is equal to ...... .

In the expansion of $(1+a)^{m+n},$ prove that coefficients of $a^{m}$ and $a^{n}$ are equal.

The value of $x$ in the expression ${[x + {x^{{{\log }_{10}}}}^{(x)}]^5}$, if the third term in the expansion is $10,00,000$

If the coefficient of $x ^7$ in $\left(a x-\frac{1}{b x^2}\right)^{13}$ and the coefficient of $x^{-5}$ in $\left(a x+\frac{1}{b x^2}\right)^{13}$ are equal, then $a^4 b^4$ is equal to :

The total number or irrational terms in the binomial expansion of $\left( {{7^{1/5}} - {3^{1/10}}} \right)^{60}$ is