Probability question A large white cube is painted red, and then cut into 27 identical smaller cubes. These smaller cubes are shuffled randomly. A blind man (who also cannot feel the paint) reassembles the small cubes into a large one. What is this probability that the outside of this large cube is completely red?
 A: These "complex" probability questions should always be addressed in a similar manner. Decompose them in smaller subproblems and solve them separately.
This problem could be decomposed in finding these probabilities:


*

*probability that the inner center is in its place

*probability that the $8$ corner pieces are at the corner positions

*probability that each corner piece is correctly orientated

*probability that the $6$ center pieces are at the center positions

*probability that each center piece is correctly orientated

*probability that the $12$ edge pieces are at the edge positions

*probability that each edge piece is correctly orientated


If you find those, you can multiply all of them and get the final result.
A: Hint: The cube has a $3 \times 3$ cube. The set of "types" is $\mathrm{S} = \{\mathrm{center, corner, edge, core}\}$. The cardinality of each of those types is:


*

*$| \mathrm{center}|= 6$, 

*$| \mathrm{corner}|= 8$,

*$| \mathrm{edge}|= 12$, and

*$| \mathrm{core}|= 1$.

