Number of positive integral solutions satisfying the equation $\left({\mathrm{x}}_1+{\mathrm{x}}_2+{\mathrm{x}}_3\right)\left({\mathrm{y}}_1+{\mathrm{y}}_2\right)=77$, is

$\begin{array}{l}\left({\mathrm{x}}_1+{\mathrm{x}}_2+{\mathrm{x}}_3\right)\left({\mathrm{y}}_1+{\mathrm{y}}_2\right)=77=11\times 7\text{ or }7\times 11\\ \mathrm{Case}-\mathrm{I}{\mathrm{x}}_1+{\mathrm{x}}_2+{\mathrm{x}}_3=11\text{ and }{\mathrm{y}}_1+{\mathrm{y}}_2=7\end{array}$
Now positive integral solution satisfying the equation are given by
$\begin{array}{l}{ }^{8+3-1}{\mathrm{C}}_{3-1}{\times }^{5+2-1}{\mathrm{C}}_{2-1}\\ {\Rightarrow }^{10}{\mathrm{C}}_2{\times }^6{\mathrm{C}}_1\\ 45\times 6=270\\ \mathrm{Case}-2{\mathrm{x}}_1+{\mathrm{x}}_2+{\mathrm{x}}_3=7\text{ and }{\mathrm{y}}_1+{\mathrm{y}}_2=11\end{array}$
Again positive integral solution satisfying the equations are given by
$\begin{array}{l}{\Rightarrow }^{4+3-1}{\mathrm{C}}_{3-1}{\times }^{9+2-1}{\mathrm{C}}_{2-1}\\ {\Rightarrow }^6{\mathrm{C}}_2{\times }^{10}{\mathrm{C}}_1\\ \Rightarrow 15\times 10=150\\ \text{ Total ways }=270+150=420\end{array}$