Life is too short to try this manually

First, we imagine smashing the Rubik's Cube (perhaps out of frustration) so we have all the individual cubes in front of us. We sort them into the following groups: corners, edges, and centers.

We'll start by focusing on the corners. It's evident that there are 8! ways to place the 8 corners at each vertex. Also note that there are 3 ways to orient each corner. We multiply 8! and 3^8 and set them to the side, calling their product α.

For edges, we follow a similar process. There are 12! ways to arrange these edges, and there are 2 different ways to orient each edges. We multiply 12! and 2^12, and set these to the side, calling their product β.

We're going to ignore the centers since they remain fixed and only have one possible orientation.

Now, the product of αβ gives us every possible arrangement we can make by smashing the cube into cublets and gluing them back together. However, not all of these arrangements are possible for a real, solvable Rubik's Cube. For example, if we change the orientation of one corner on a solvable cube, we'll get an unsolvable cube. There are 12 little "universes" of cubes we can have and 11 of them are unsolvable. Thus, (1/12)(αβ) gives us a large number that is the total number of arrangements of a Rubik's Cube. It is a number, unfortunately, which I don't want to write all the way out again because of my chronic laziness.

I am a speedcuber, (average solve : 19 secs )

The possible permutations ( by rotating the faces alone ) can be calculated by considering the 8 corners and 12 edges . The center faces are fixed. The possible permutations for the corners is 8! . The possible orientations for the 7 corners are independent of each other. But the last corner's orientation depends on the previous corner. This is because a single corner or an edge cannot be independently oriented without disturbing the other. So, there are $3^{7}$ possible orientations for the 8! permutations. Now the 12 edges can be permuted in $\frac {12!} {2}$ ways ( even permutation ). Similarly to corners, the 11 edges have 2 orientations independent to each other while the 12th depends on the previous one. Hence there are $2^{11}$ possible orientations.

Thus combining all the combinations, we get,

8!\quad \times \quad { 3 }^{ 7 }\quad \times \quad \frac { 12! }{ 2 } \times \quad { 2 }^{ 11 }\quad =\quad 43,252,003,274,489,586,000 .

That is 43 Quintillion ( $43\times 10^{ 18 }$ ) . This is just the combinations obtained by rotating the faces alone. One can dissemble the cubies and create new so called Universes ( in the Cube world) to have even more combinations.

That would be

8!\quad \times \quad { 3 }^{ 8 }\quad \times \quad 12!\quad \times \quad { 2 }^{ 12 }\quad =\quad 519,024,039,293,878,272,000

Calculated values from : Wikipedia