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The total number of positive integral solutions for $(x, y, z)$ such that $x y z = 24 .$

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Detailed Solution

$∵ x y z = 24 = 2^{3} \times 3^{1}$

Hence, total number of positive integral solutions

$\begin{array}{l} = (^{3 + 3 - 1} C_{3 - 1}) (^{1 + 3 - 1} C_{3 - 1}) \\ =^{5} C_{2} \times^{3} C_{2} = 30 \end{array}$

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