2005 AIME I Problem 9

Twenty-seven unit cubes are 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^br^c}, where p,q, and r are distinct primes and a,b, and c are positive integers, find a+b+c+p+q+r.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.