If you were to calculate the number of possible combinations on a cube mathematically then you would assume it would be (2^12)*(2!)*(3^8)*(8!) which is

This is because the cube has 12 edges and 8 corners, the 12 edges can each be put in 12 different locations and flipped one of two different ways. The 8 corners can be put in 8 different locations and twisted in 3 different ways. However, this number assumes that every single case can be reached which is not actually the case.

There are a few cases which cannot be reached with normal legal moves. These cases can be reached by randomly reassembling the cube but not by doing normal legal moves.

The most common impossible case is a single twisted corner, it is not possible to twist only one corner with normal moves: 

If you were to add up all of the twists done on the cube, it must be a multiple of 3 for the cube to be solveable, so two corners twisted in the same direction is not possible either (but can be reduced down to one twisted corner), while two corners twisted in opposite directions is possible. One challenge in cube hardware design over the past 15 years has been regarding how to make a cube turn fast and smooth without the corners twisting by accident, there are famous incidents such as Feliks Zemdegs 5.33 solve in 2015 which would have been a world record at the time if his cube didn't get a corner twist right at the very end of the solve. Modern cubes reduce the risk of corner twists using squared off corners, where the corner stalk is large and hooks properly onto the edges, however, cube designers often face a trade off where this may slow the cube down. Cubers may consider it worth having a corner twist every 100 solves if the cube is faster and performs better otherwise.

If your cube has a corner twist then you need to twist it back to solve it, this is pretty easy to do.

The next impossible case is that of the edge flip, if this happens then it means that the cube has been disassembled at some point as it is not possible to flip an edge on a cube without taking the edge out and putting it back the wrong way. 

There can never be an odd number of flipped edges left on the cube after turning the cube with normal legal moves and the only way to solve the case of 1 flipped edge as shown in the image is to take the edge out and put it back in the right way up. There is a technique behind taking edges out and putting them back in, if you are new to it then I would recommend loosening the screws in the centres around that edge (just remember to tighten them back up again afterwards). To take the edge out, you need to move the middle slice up 45° and then grab on to the edge with your thumb and 1 finger and twist the edge piece until it comes out, gently pulling it too if necessary. To put it back in, you need to separate the two corners next to it slightly, then put the edge in at a 40° angle, push and twist.

The final impossible case is the case of two swapped edges, you can never be left with just two edges swapped, it is possible however, to have two edges and two corners swapped at the same time. If all the corners are solved, then you can either have 3 corners to cycle (U perm), opposite swap on two sets of two edges (H perm) or diagonal swap on two sets of two edges (Z perm). 

Obviously, as with the other images, we are assuming that all the other sides of the cube are fully solved. The only way to solve this case is to take the two edges out of the cube and put them back in the right way up. This can be done the same way as the flipped edge case.

So, if you were to randomly reassemble a completely dismantled cube, what are the chances that it would be possible to be solved?

The answer is actually just 1 in 12 or 8.33333...% and that leaves the true number of possible combinations at the well known number of , this is still far more combinations than will ever be seen by humanity, if you were to do 20 random moves on your cube right now, then there is probably less than a 1 in a million chance that your scramble has ever been on a cube before.

It is important to be careful when dismantling cubes as it is possible to generate a case which can never be solved without dismantling it again. If you are new to cubing then you will probably at some point have a more experienced cuber tease you by giving you a cube with an impossible case, but if you know this stuff then you can outsmart them when it happens by immediately recognising when you get to last layer that the cube has been tampered with.

If you get to last layer and see that only one edge is unsolved:

What the experienced cuber teasing you wants to see is you doing F R U R' U' F' over and over again, as you get really confused with why your algorithm isn't working. But now you know that the case is not possible, you can just calmly take out the edge and fix it.

Likewise, with the corner twist, this is less likely to be deliberate but it can happen:

You can now recognise that both corners shouldn't be twisted in the same direction and simply doing R U R' U R U2' R' over and over again will never solve it, neither will any more advanced 2 look OLL algorithm. So you can simply twist the corner and continue as normal.