The total number of positive integral solutions x ,   y ,   z such that x y z = 24 is :

We have,

$xyz=24$

$\Rightarrow xyz=2^3\times 3^1$

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,

${\alpha }_1+{\alpha }_2+{\alpha }_3=3 ...\left(\mathrm{i}\right)$

${\beta }_1+{\beta }_2+{\beta }_3=1 ....\left(\mathrm{ii}\right)$

Number of non-negative integral solution for $\left(\mathrm{i}\right)$

$={}^{5}C_2=10$ 

Number of non-negative integral solution for $\left(\mathrm{ii}\right)$

 $={}^{3}C_2=3$

Total number of positive integral solutions $\left(x,y,z\right)$ such that $xyz=24$ is

$=10\times 3=30$