Explanation

(a)
Coefficient of $x^{7}$ in ${(1 + 3x - {2x}^{3})}^{10}$
$= \sum_{}^{}{\frac{10!}{n_{1}!n_{2}!n_{3}!}{(1)}^{n_{1}}{(3)}^{n_{2}}{( - 2)}^{n_{3}}}$
Where, $n_{1} + n_{2} + n_{3} = 10,\ \ \ n_{2} + {3n}_{3} = 7$
Different possibilities are as follows
$\begin{matrix}
n_{1} & n_{2} & n_{3}
\end{matrix}$
$\begin{matrix}
3 & \ \ 7 & \ \ 0 \\
5 & \ \ 4 & \ \ 1 \\
7 & \ \ 1 & \ \ 2
\end{matrix}$
$\therefore$ Coefficient of $x^{7} = \frac{10!}{3!7!}{(1)}^{3}{(3)}^{7}{( - 2)}^{0}$
$+ \frac{10!}{5!\ 4!\ 1!}{(1)}^{5}{(3)}^{4}{( - 2)}^{1} + \frac{10!}{7!\ 1!\ 2!\ }{(1)}^{7}{(3)}^{1}{( - 2)}^{2}$
$= 62640$