The total number of positive integral solutions $(x, y, z)$ of $x y z=24$ is
The total number of positive integral solutions $(x, y, z)$ of $x y z=24$ is
$24$
$30$
$36$
$32$
Solution
$x y z=24=2^3 \times 3$
Let $x=2^{x_1} \times 3^{y_1}$,
$y=2^{x_2} \times 3^{y_2}$ and $z=2^{x_3} \times 3^{y_3}$
$x_1+x_2+x_3=3$ and $y_1+y_2+y_3=1$
Number of non-negative integral solution
$={ }^{3+3-1} C_{3-1}={ }^5 C_2=10$
Number of non-negative integral solution
$={ }^{1+3-1} C_{3-1}={ }^3 C_2=3$
$\therefore$ The total number of positive integral $10 \times 3=30$