If you have read my response to an earlier question, then you know that you are using a model in which objects-distinguishable, cells-distinguishable and cells may not be empty. Putting R distinguishable balls into N distinguishable urns without any urn being empty of course requires . The number of ways to do that is simply the number of surjections (onto functions) from a set of R elements to a set of N elements. I will give you the counting formula for that: .
To understand this idea, you may want to study Stirling numbers of the second kind.