Let 'a' be a factor of 120, then the number of  positive integral solutions of ${\mathrm{x}}_1{\mathrm{x}}_2{\mathrm{x}}_3=\mathrm{a}$ is 

$x_1{x}_2{x}_3{x}_4=120$ where $120/a=x_4$
                $=2^3\cdot 3\cdot 5$
For 2³, the no. of ways ${=}^6{C}_3=20$ 
For 3, the no. of ways ${=}^4{C}_3=4$ 
Similarly, for 5, the no. of ways${=}^4{C}_3=4$
Total number of integral solutions = 20 x 4x 4 = 320