2005 AIME I Problem 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.