Note that $x! + y + z \ge 3$ for any positive integers $x,y,z$. So all three expressions have to be odd. Add all of them we conclude that $a! + b! + c!$ is odd, i.e. one of them has to be $1$. Wlog it is $a$. Then $b! + c + 1$ and $c! + b + 1$ has to be prime. We can easily conclude that $b = c$ or otherwise WLOG $b < c$ then $b + 1 \mid c!$. Now, the question asks to prove $b + 1$ is a prime. Else, there exists $p \mid b + 1$ such that $p \le b$ and therefore $p \mid b! + b + 1$, contradiction.