Factorial and a Power of Two

$\large a! b!=a!+b!+2^c$

Let all the triplets of positive integer solutions $(a,b,c)$ satisfying the equation above be $(a_1, b_1, c_1) , (a_2, b_2, c_2) , \ldots , (a_n , b_n, c_n)$ . Find $(a_1 + b_1 + c_1) + (a_2 + b_2+c_2) + \cdots + (a_n + b_n + c_n) .$

$$ Notation :
$!$ denotes the factorial . For example, $8! = 1\times2\times3\times\cdots\times8$ .