Painting A Cube

Lets start with base version of this problem. Assume you want to paint a cube using 6 different colors. How many ways can we do this? This basic problem is easy to see that you can do it in $6!$ ways ( $6$ choices for one side, $5$ for the next, then $4$ , then...).

Now that we have a base understanding of $6!$ or $720$ different ways to paint a cube, now we have to filter out the ones that are the same but with a different orientation. A cube resting on a table (maintaining the same side as "up" and "down") has $4$ different orientations; each side facing "forward". Since there are $6$ sides, $4$ orientations each, there are $24$ ways to orient a cube. So we take our $720$ painted cubes, knowing $24$ are the same cube oriented differently and we get $720 / 24 = 30$ .

Suppose the colors are red, orange, yellow, green, blue and purple.

Without loss of generality you can paint one side green. Now you have five choices of which color to choose for the opposite side. For each of those five choices, without loss of generality you can pick a side for any third color you choose, leaving $3$ sides unpainted. Every combination for painting these remaining sides leads to a different final outcome. Therefore, once you pick the opposite color (for which there are five choices) you have $3! = 6$ unique coloring patterns to choose from.

So, the total number of ways is $5*3! = \boxed{30}$ ways.

In cube, there are always $3$ distinct pairs of opposite faces.

From set of $6$ elements we can choose $3$ distinct pairs of elements in $\dfrac{6!}{2!^3 \cdot 3!} = 15$ ways.
( $6!$ which is all permutations divided by $2!$ three times since order in pair doesn't matter and also by $3!$ because order of pairs doesn't matter).

Hence, there are $15$ ways to choose distinct pairs of opposite faces.
However, for each cube, there exists a mirror image of that cube which has the same pairs of opposite faces but it is not the same cube rotation-wise.

Hence, there are $2\cdot15=\boxed{30}$ ways to color a cube in such way.

I will use numbers 1 to 6 instead of 4 different colours. Since blank/plane dice i.e, cube is symmetric

So,first assign number 1 to any side of blank dice to make reference (say this side as S1). This can be done in only one way.

Now selecting one number from 5 numbers (i.e, 2,3,4,5,6) select any one to assign selected number to opposite side of S1, this can bee done in 5 ways (since there are five numbers left)

And , now put that dice (which has 4 blank sides) by your two fingers in such manner that those 4 blank side can rotate in circular manner

Now this four blank can be assigned by four left numbers in 3!=6 ways (circular arrangement)

So total number of ways.. C(5,1) 3!=(5) 3!=☆(30)☆

Let's say that 6 different colors are c1, c2, c3.. c6.

Now let's consider that the face of cube facing up is painted with c1. Then the face of cube at the bottom can be painted in 5 different ways.

And the 4 faces on the horizontal side are in circular permutation so they can be painted in (4–1)! ways

So total number of ways cube can be painted

= 5* (4–1)! = 5* 3! = 5* 3* 2 = 30 ways