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 $$\frac{p^a}{q^br^c},$$ where $$p,q,$$ and $$r$$ are distinct primes and $$a,b,$$ and $$c$$ are positive integers, find $$a+b+c+p+q+r.$$

(第二十三届AIME1 2005 第9题)