The total number of positive integral solutions (x, y, z) such that...

We have,

$xyz = 24$

$⇒ xyz = {2}^{3} \times {3}^{1}$

Let

$x = {2}^{{\alpha}_{1}} \times {3}^{{\beta}_{1}}$

$y = {2}^{{\alpha}_{2}} \times {3}^{{\beta}_{2}}$

$z = {2}^{{\alpha}_{3}} \times {3}^{{\beta}_{2}}$

where,

${\alpha}_{1} + {\alpha}_{2} + {\alpha}_{3} = 3 . . . \left(i\right)</p><p>${\beta}_{1} + {\beta}_{2} + {\beta}_{3} = 1 . . . . \left(ii\right)$</p><p>Number of non-negative integral solution for \left(i\right)$

$= C2 5 = 10$

Number of non-negative integral solution for \left(ii\right)$</p><p>$= C2 3 = 3$</p><p>Total number of positive integral solutions \left(x, y, z\right)$ such that $xyz = 24$ is

$= 10 \times 3 = 30$