cool variables as hot constants !

It is Good Application of AM-GM.

Since We have 4 variables and 2 equations So it can't solved directly. And all 4 variables are positive real variables. So first Thing that must strike in our Mind is that is AM-GM inequality can be used or not ?

( I mean Is there is boundedness of cool variables or not ? )

Let's Take a try:

$\because \quad a+b+c+d=12\\ \therefore \quad \frac { a+b+c+d }{ 4 } \quad \ge \quad { (abcd) }^{ \frac { 1 }{ 4 } }\\ { \therefore \quad (abcd) }_{ max }=81$ .

Now look to the second Information:

$\because \quad a.b.c.d=27+a.b+a.c+a.d+b.c+b.d+c.d\\ LHS\quad \le \quad 81\quad \quad \longrightarrow (1)$ .

Similarly By using AM-GM inequality on variables Of RHS

$RHS\quad \ge \quad 27\quad +6(\sqrt [ \frac { 3 }{ 4 } ]{ abcd } )\\ \\ RHS\quad \ge \quad 81\quad \quad \longrightarrow (2)$ .

But LHS=RHS=81 only possible solution.

So AM=GM and therefore numbers are equal .

$a=b=c=d=3$ .

Now Put the Value and get the Answer.

It is interesting that we have given 4 variables and 2 equation's but still they can be solved By using simple AM-GM inequality.