Painting a cube
Six colors of paint are available. Each face of a cube is to be painted a different color. In how many different ways can this be done if two colorings are considered the same when one can be obtained from the other by rotating the cube?
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1 solution
Let us paint one of the faces of the cube as Red (one of the colours). Now we can colour the face opposite to this in 5 ways. Now, we have faces in between to paint. These four faces are at the moment not distinguishable from each other. Let us colour one of the faces. This can be done in 1 way. Now, the other 3 faces will be distinguishable . These 3 faces can be painted in 3 ! = 6 ways. Hence, total ways = 5 *6 =30 ways.