General Term and Middle Term

Coefficient of ${\alpha }^t$ in the expansion of ${\left(\alpha +p\right)}^{m-1}+{\left(\alpha +p\right)}^{m-2}\left(\alpha +q\right)+{\left(\alpha +p\right)}^{m-3}{\left(\alpha +q\right)}^2+...{\left(\alpha +q\right)}^{m-1}$, where $\alpha \ne -q$ and $p\ne q$, is:

The sum $\frac{1}{1!2004!}+\frac{1}{3!2002!}+\frac{1}{5!2000!}+\dots +\frac{1}{2003!2!}=\left(\frac{2^a-1}{ b!}\right)$ where $a$ and $b$ are positive integers. Then
find $b-a.$

In the expansion of ${\left(3^{\frac{-x}{4}}+3^{\frac{5x}{4}}\right)}^n$, the sum of the binomial coefficients is $256$ and four times the term with greatest binomial coefficient exceeds the square of the third term by $21n$, then find $4x$.