If  $2a+3b+4c=\gamma$, then for some  $\gamma$ the family of line  $ax+by+c=3$ always passes through a fixed point  $(\alpha ,\beta )$, then the value of $\left[\alpha \beta \gamma \right]$  is (where  $[.]$ denotes the greatest integer function)

Equation of line $\frac{ax}{c-3}+\frac{ by }{c-3}+1=0$  has two independent parameters. Now there must be one linear relation between $\frac{a}{c-3}$  and $\frac{b}{c-3}$  which is independent of  $a,b$ and  $c$ given relation can be expressed as  $\frac{2a}{c-3}+\frac{3b}{c-3}=\frac{\gamma -4c}{c-3}$
Now R.H.S is independent of  $\gamma$ if $\gamma =12$ and for  $\gamma =12$, line passes through a fixed point    $(\frac{1}{2},\frac{3}{4})$
$\Rightarrow \left[\alpha \beta \gamma \right]=4$