# how to evaluate combinations of different configurarations of a rubik's cube

1. how to calculate the total combination number of an rubik's cube ?

2. and what is the number of combination if we fix one face of it with all matched colours , specifically saying blue one ,??

3. no. of combination of two adjacent faces , say blue and yellow are with their matched colours ( ,if it is possible )

4. no. of combination with 3 adjacent faces are matched with their colours (say the faces are red,blue and yellow ,, again if it is possible)

5. same thing with 4 adjacent faces ,say blue yellow ,, red and white ,are matched (again if it is possible )

1. There are 8 corner cubies, each with 3 orientations; 12 edge cubies, each with 2 orientations; you cannot rotate a single corner cubie; you cannot flip a single edge cubie; you cannot have odd number of permutations. This leads to: $$\frac{8!3^812!2^12}{3\cdot2\cdot2}\approx4.3\times 10^{19}.$$
2. 4 corner cubies and 8 edge cubies are flexible, so it is: $$\frac{4!3^48!2^8}{3\cdot2\cdot2}\approx1.7\times 10^9.$$
3. 2 corner cubies and 5 edge cubies are flexible, so it is: $$\frac{2!3^25!2^5}{3\cdot2\cdot2}=5760.$$
4. 1 corner cubies and 3 edge cubies are flexible, so it is: $$\frac{1!3^13!2^3}{3\cdot2\cdot2}=12.$$
5. Only one edge cubie is not matched, so it cannot be flipped. There is only $1$ possible arrangement.