2005 AIME II Problems/Problem 9

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Twenty seven unit cubes are painted orange on a set of four faces so that two non-painted faces share an edge. The $27$ cubes are randomly arranged to form a $3\times 3\times 3$ cube. Given the probability of the entire surface area 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 $a+b+c+p+q+r$.


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