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Question

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


A

36

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B

45

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C

24

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D

30

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Solution

The correct option is D

30


Explanation of the correct option.

Given : xyz=24

It can be written as x.y.z=23.31

Thus,

x=2a1.3b1y=2a2.3b2z=2a3.3b3

We can see that a1,a2,a30,1 and b1,b2,b30,1

Since, a1+a2+a3=3

Thus, non negative solution=C(3-1)(3+3-1)

=C25=5!2!(5-3)!=10

Since, b1+b2+b3=1
Thus, non negative solution=C(3-1)(1+3-1)

=C23=3!2!(3-2)!=3

Therefore, total solutions =10×3

=30

Hence OptionD is the correct option.


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