Question 9
Twenty-seven unit cubes are each painted orange on a set of four faces so that the two unpainted faces share an edge. The $27$ cubes are then randomly arranged to form a $3 \times 3 \times 3$ cube. Given that the probability that the entire surface of the larger cube is orange is $\frac{p^a}{q^b r^c},$ where $p,$ $q,$ and $r$ are distinct primes and $a,$ $b,$ and $c$ are positive integers, find $a + b + c + p + q + r.$