I think for the cuboid the ways of painting should be 360
While finding the number of ways to paint a cube, why didn’t we consider a factor of 6 ways for selecting the first colour (red in the given example)?
We are assuming a cube freely floating in space (i.e. i can turn/twist it in any direction). So when unpainted, all the faces will be effectively identical. So since every colour has to appear once, I just arbitrarily chose a colour and a position for it and then decided where to place the other colours relative to it. You could have started with any colour, it would make no difference.
In the cube case,after painting first two opposite sides,the remaining four sides can be viewed as four places in a circle.So they can be arranged in 3! ways.So total no of ways is 5*3!=30 ways.
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