Though this question can be done better the way already discussed, Let's count the number of cubes with 1-face, 2-faces and 3-faces painted. This might give us a better picture if the question is modifies as : probability of getting a cube with exactly 2 faces painted or exactly 3 faces painted.

1-face painted:Cubes with one face painted lies on each face except along the edges of the face.

So total number of 1-face painted cubes = 8*8*6 = 384<when we exclude edges, we are left with 8*8><6 for 6 faces>

2-faces painted: Such cubes exist along the edges except the ones at the corners. And total number of edges = 12

So, total number of 2-faces painted cubes = 8*12 = 96 <Out of 10 cubes along an edge, only 8 are 2-faces painted, the ones at the corners are all three faces painted>

3-faces painted:The ones at the corners are 3-faces painted. Number of corners = 8

Total number of 3-faces painted cubes= 8.

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