In how many different ways can a cube be painted by using N different colors of paint?
Note that this question is not same to Painting the faces of a cube with distinct colours as the colours here may not be distinct (i.e. used many times). For Painting the faces of a cube with distinct colours, we must have $N>=6$, and the result will be:
$$\binom{n}{6}(5)(3!)$$
Can anyone help how to tackle such problem?
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$\begingroup$See here:
Specifically, under Examples -> Colored cubes.
$\endgroup$ $\begingroup$I'm not sure if the answer is correct, but here's some of my idea: if we are painting a cube with 6 distinct colours. fixing one of the faces, the face on the opposite side has 5 colour choices using the circular arrangement for the rest of the 4 sides we have 3! ways. In total, we have 5*6=30 ways.
Now if we have n colours, after fixing the first face the opposite face has (n-1) colour choices. There are now (n-2) colours left to choose and for the four faces left we might choose 4 colours from the (n-2) colours and do the circular arrangements again.
In total, we have (n-1)*[(n-2)C4]*3! ways.
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