cube-with-'n'-cuts problem

Just a while before, I was thinking of some problem which very queerly reminded of another problem we used to tackle, while solving those mental ability questions during NTSE preparation in X standard. It's not anything special or hard, and I am pretty sure this result I've derived here would be present in an umpteen no. of books/guides; but maybe it's still nice to put it on the blogpage so that if someone is searching for a quick solution, then I believe the formalism I present here will make it easy to remember.

So the problem is - you take a cube and uniformly paint it all around, then make n equally-spaced cuts along each of the x, y and z axes. This gives you n³ smaller cubes and one needs to determine how many cubes have exactly one face coloured, or two faces coloured etc. etc.

The following formula contains the answers:

n³ = (n - 2 + 2)³
= (n - 2)³ + 6(n - 2)² + 12(n - 2) + 8

The first term is the number of cubes with none of the faces coloured, second is with exactly one coloured face and so on.... and this is pretty easy to visualize too.

The interesting thing now is, can this be generalized to higher dimensions? As in, would cutting a kube (a k dimensional cube :-P) with n cuts in each of the k dimensions (to obtain n^k kubes in all i.e.) result in q = ⁿC_k (n-2)^j 2^j being the no. of kubes with exactly j faces coloured.