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The total number of positive integral solutions $\mathrm{(a, b, c)}$ such that $\mathrm{a b c=48}$ is
Option: 1
45

Option: 2
30

Option: 3
65

Option: 4
90

Answers (1)

It is given that $\mathrm{a b c=48}$
Then it can also be written in the form $\mathrm{a . b \cdot c=2^{4} \cdot 3^{1}}$

Then a,b,c can be written in the form

$\mathrm{a=2^{a_{1}} \cdot 3^{b_{1}} }$
$\mathrm{b=2^{a_{2}} \cdot 3^{b_{2}} }$
$\mathrm{c=2^{a_{3}} \cdot 3^{b_{3}} }$

$\mathrm{\left(a_{1}, a_{2}, a_{3}\right) \varepsilon(0,1) }$ and $\mathrm{\left(b_{1}, b_{2}, b_{3}\right) \varepsilon(0,1) }$
Also $\mathrm{a_{1}+a_{2}+a_{3}=4 }$

Then the number of solutions will be $\mathrm{={ }^{4+3-1} C_{3-1} }$
                                                           $\mathrm{=^{6} C_{2}}$
                                                           $\mathrm{=15}$

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 $\mathrm{=15 \times 3}$
                                                          $\mathrm{=45}$

The correct option is (a).