Painting a cube using N different colors? 
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?

See here:  http://en.wikipedia.org/wiki/P%C3%B3lya_enumeration_theorem
Specifically, under Examples -> Colored cubes.
