Number of positive integral solutions of $x y z = 21600$ is

1320

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4620

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1260

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205

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Solution

The correct option is C $1260$
$x y z = 21600$
$x y z = 2^{5} \times 5^{2} \times 3^{3}$
$x y z = 2^{a_{1} + a_{2} + a_{3}} 3^{b_{1} + b_{2} + b_{3}} 5^{c_{1} + c_{2} + c_{3}}$
$\Rightarrow a_{1} + a_{2} + a_{3} = 5$
$\Rightarrow b_{1} + b_{2} + b_{3} = 2$
$\Rightarrow c_{1} + c_{2} + c_{3} = 3$
No. of solution of $1 s t$ $e q^{n}$ is $^{n + r - 1} C_{r - 1} =^{7} C_{2} = 21$
$\Rightarrow 2 n d^{4} C_{2} = 6$
$\Rightarrow 3 r d^{5} C_{2} = 10$
Total solution, $21 \times 6 \times 10 = 1260.$
Hence the answer is $1260.$

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