If 3 sides of a rubik's cube is solved, how many possible combinations are there in total?

Question

If we have a cube like this, how many possible combinations of the other 3 sides are there in total?

Answer 1

I'm assuming you count only positions that can be reached from a solved cube.

We can see 7 corner cubies at least partially. The ones where we can see at least two sides must be in the right position and orientation, and that means that the ones where we can see one side are also in the right position and orientation, because they carry the only remaining corner sticker in the each of the colors we can see.

This means that the eighth corner cubies is also in the right place, and since the orientation of all the visible corners are right, the last corner is also in the right orientation. So the corners are completely solved.

For edges we can see three in the correct place and orientation, and three pairs of two that each may be swapped or not.

The three edges we can't see may be permuted in $3!=6$ ways, and depending on whether that permutation is odd or even, either (0 or 2) or (1 or 3) of the pairs of half-visible edges must be swapped -- in each case there are $1+3=4$ possibilities for that.

Now the orientation of two of the hidden edges can be chosen freely, but the third is then determined. This gives $2^2=4$ possibilities.

Combining all that, we get $$ 3!\cdot (1+3) \cdot 2^2 = 96 $$ reachable states of the cube that have these three sides solved.