# Factorial and a Power of Two

Number Theory Level 5

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

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, $$8! = 1\times2\times3\times\cdots\times8$$.

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