$\text{The sum of the digits of the number of positive integral solutions satisfying the equation} \left(x_1+x_2+x_3\right)\left(y_1+y_2\right)=77$

$\left(x_1+x_2+x_3\right)\left(y_1+y_2\right)=11\times 7\text{ or }7\times 11$

$\text{ In the first case, }\left(x_1+x_2+x_3\right)=11\text{ and }\left(y_1+y_2\right)=7\text{, which have }$

${ }^{10}{C}_2{\cdot }^6{C}_1\text{ solutions }$

$\text{ In the second case, }\left(x_1+x_2+x_3\right)=7\text{ and }\left(y_1+y_2\right)=11\text{, which }$

${\text{ have }}^6{\mathrm{C}}_2{\cdot }^{10}{\mathrm{C}}_1\text{ solutions }$

$\therefore \text{ Total number of solutions }{=}^{10}{C}_2{\cdot }^6{C}_1{+}^6{C}_2{\cdot }^{10}{C}_1$

                                                        $=270+150=420$