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

36

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45

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24

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30

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Solution

The correct option is D $30$
$x . y . z = 24$
$x . y . z = 2^{3} {.3}^{1}$
$x = 2^{a_{1}} \cdot 3^{b_{1}}, y = 2^{a_{2}} \cdot 3^{b_{2}}, z = 2^{a_{3}} \cdot 3^{b_{3}}$
$a_{1}, a_{2}, a_{3} \in {0, 1, 2, 3}$
$b_{1}, b_{2}, b_{3} \in {0, 1}$
Case: $1$ $a_{1} + a_{2} + a_{3} = 3$
Non negative solution $=^{3 + 3 - 1} C_{3 - 1} =^{5} C_{2} = 10$
Case: $2$ $b_{1} + b_{2} + b_{3} = 1$
Non negative solution $=^{1 + 3 - 1} C_{3 - 1} =^{3} C_{2} = 3$
$∴$ Total solutions $= 10 \times 3 = 30$

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