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The total number of positive integral solutions \mathrm{(a, b, c)} such that \mathrm{p q r=96} is

Option: 1

 42


Option: 2

56


Option: 3

72


Option: 4

63


Answers (1)

best_answer

It is given that \mathrm{p q r=96}

Then it can also be written in the form \mathrm{p q r=2^{5} \cdot 3^{1}}
Then \mathrm{p, q}, and \mathrm{r} can be written in the form

\mathrm{p=2^{a_{1}} 3^{b_{1}}}
\mathrm{ q=2^{a_{2}} 3^{b_{2}}}
\mathrm{ r=2^{a_{3}} 3^{b_{3}}}

\mathrm{ \left(a_{1}, a_{2}, a_{3}\right) \varepsilon(0,1) \text { and }\left(b_{1}, b_{2}, b_{3}\right) \varepsilon(0,1)}

Also \mathrm{ a_{1}+a_{2}+a_{3}=5}
Then the number of solutions will be \mathrm{ ={ }^{5+3-1} C_{3-1}}
                                                           \mathrm{ ={ }^{7} C_{2}}
                                                           \mathrm{ =21}

and \mathrm{ b_{1}+b_{2}+b_{3}=1} 

Then the number of solutions will be given as \mathrm{={ }^{(1+3-1)} C_{(3-1)}}
                                                                         \mathrm{={ }^{3} C_{2}}
                                                                         \mathrm{=3}

The total number of solutions will be \mathrm{=21 \times 3}
                                                          \mathrm{=63}

The correct option is (d).

Posted by

SANGALDEEP SINGH

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