Question

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

A
36
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B
90
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C
120
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D
none of these
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Solution

## The correct option is C 120The number 24 can be factored as: 24=23×3. Hence, you need x,y,z to be of the form 2a3b You have 3 powers of 2 and one power of 3 which you need to distribute between three numbers.For 2′s, this can be done in 5C2=10 ways(a1+a2+a3=3 =>n+r−1Cr−1), and for 3 in 3 ways(b1+b2+b3=1). Hence, the answer is 3×10=30.The above assumes you are looking only for positive solutions. If x,y,z are allowed to be negative, the solutions are either the ones found above (there are 30 of these), or the ones found above with some two variables multiplied by −1 (there are 90 of these). In total, we get 120 solutions.Hence option C is correct

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