Solution to the Cube Painting problem

There r " 30 " ways in which the cubes can be painted..

Before saying the solution, lemme say why the answer is not " 1 ".. The most common answer i got from my friends is "1". If there was just 1 way to paint the cube then if u have many cubes of this type they would be indistinguishable from each other.. but this is not the case here. A cube with red and blue painted on opposite sides can be distinguished from a cube with red and green painted on opposite sides..

This gives us a hint to the solution.. the cubes can distinguished by what colors lie opposite to each other. So we just have to consider the number of ways to make such pairs..

Now consider an unpainted cube.. Now paint a particular side with 1 color.. Now u have 5 colors left. The number of ways in which the side opposite to this can be colored is in 5 ways.. After coloring the opposite side u have 4 colors left.. Rotate the cube by 90.. and color the blank face wit one more color. Then the face opposite to it an be colored in 3 ways.. Now 2 faces are left.. Color the remaining 2 colors on the 2 sides.. But we will be able to make 1 more distinguishable cube if u jst reverse the way in which the last 2 colors are painted. So the number of cubes tat can be made is 5 * 3 *2 = 30 cubes..