In the expansion of ${\left(\sqrt[4]{{\mathrm{x}}^{-3}}+\mathrm{a}\sqrt[4]{{\mathrm{x}}^5}\right)}^{\mathrm{n}}$, the sum of all binomial coefficients lies between 200 and 400 and the term independent of x is 448. Then the value of a is

$$\begin{gathered} {\left({\mathrm{x}}^{\raisebox{1ex}{$-3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}+\mathrm{a} {\mathrm{x}}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\right)}^{\mathrm{n}} \\ 200<2^{\mathrm{n}}<400, \Rightarrow \mathrm{n}=8 \end{gathered}$$

For Independent of $\left(\mathrm{a} {\mathrm{x}}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}+{\frac{1}{{\mathrm{x}}^{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}}}^8\right)$

$\mathrm{f}=\frac{\mathrm{nP}}{\mathrm{p}+\mathrm{q}}=\frac{\left(8\right) \left({\displaystyle \raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\right)}{{\displaystyle \frac{5}{4}}+{\displaystyle \frac{3}{4}}}=\frac{10}{2}=5$

$$\begin{gathered} {}^8{\mathrm{C}}_5 {\left(\mathrm{a}\right)}^3=448 \Rightarrow 56{\mathrm{a}}^3=446 \\ ={\mathrm{a}}^3=8 \\ =\mathrm{a}=2 \end{gathered}$$