\((x_1 + x_2 + x_3)(y_1 + y_2) = 77=7*11\)
We are looking for positive integral solutions, meaning none of them will be 0 or negative.

So two cases can be

1) \((x_1 + x_2 + x_3)=7\), that is we have to find ways to distribute 7 things in three parts => (7-1)C(3-1)=6C2=15
\((y_1 + y_2) = 11\), => (11-1)C(2-1)=10C1=10
Total ways = 15*10=150

2) \((x_1 + x_2 + x_3)=11\), that is we have to find ways to distribute 11 things in three parts => (11-1)C(3-1)=10C2=45
\((y_1 + y_2) = 7\), => (7-1)C(2-1)=6C1=6
Total ways = 45*6=270

Total ways = 150+270=420.


You can also find solutions separately for each equation but it will be time consuming
\((y_1 + y_2) = 11\)
Now possibilities are (10,1); (9,2) and so on
Also \((x_1 + x_2+x_3) = 11\)
Here possibilities will be 9-1-1 and ways to fill up 3!/2! and so on.