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 ${(1 + x + {x^3} + {x^4})^{10}},$ the coefficient of ${x^4}$ is

Find an approximation of $(0.99)^{5}$ using the first three terms of its expansion.

A ratio of the $5^{th}$ term from the beginning to the $5^{th}$ term from the end in the binomial expansion of $\left( {{2^{1/3}} + \frac{1}{{2{{\left( 3 \right)}^{1/3}}}}} \right)^{10}$ is

If $\left(\frac{3^{6}}{4^{4}}\right) \mathrm{k}$ is the term, independent of $\mathrm{x}$, in the binomial expansion of $\left(\frac{\mathrm{x}}{4}-\frac{12}{\mathrm{x}^{2}}\right)^{12}$, then $\mathrm{k}$ is equal to ...... .

If the coefficient of the second, third and fourth terms in the expansion of ${(1 + x)^n}$ are in $A.P.$, then $n$ is equal to