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The total number of positive integral solutions \mathrm{(a, b, c)} such that \mathrm{p q r=192} is

Option: 1

84


Option: 2

72


Option: 3

64


Option: 4

 56


Answers (1)

best_answer

It is given that \mathrm{p q r=192}

Then it can also be written in the form \mathrm{p q r=2^{6} \cdot 3^{1}}
Then \mathrm{p, q, 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)} and \mathrm{\left(b_{1}, b_{2}, b_{3}\right) \varepsilon(0,1)}

also \mathrm{a_{1}+a_{2}+a_{3}=6}
Then the number of solutions will be \mathrm{={ }^{6+3-1} C_{3-1}}
                                                          \mathrm{= { }^{8} C_{2}}
                                                          \mathrm{=7 \times 4}
                                                          \mathrm{=28}

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{=28 \times 3}
                                                          \mathrm{=84}

The correct option is (a).

 

Posted by

Pankaj Sanodiya

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