Given six distinct colors, in how many unique ways can a six faced cube be painted such that no two faces have the same color?
Note: Mixing of colors is not allowed.
Answer: 30 ways
To avoid repetitions, let us fix the color of the top face.
Hence, the bottom face can be painted in 5 ways.
Now, what remains is the circular arrangement of the remaining four colors which can be done in (4-1)! = 3! = 6 ways (For n distinct objects, number of distinct circular arrangements are (n-1)!).
Hence, the answer is 5*6 = 30 ways.