Combinatorics/counting: How many configurations are possible for m differenct objects in n boxes of unlimited occupany (m<n)

Hello! The question is pretty much entirely in the title. I wasn't sure if I could just use the fundmental principle of counting here with m^n, as when I look up related formulae in statistical mechanics I get a different result.
Just to be clear:
Say there are 5 distinguishable objects and 20 distinguishable boxes. I can put up to 5 objects in a box. I am interested in how many different configurations, or states, are possible under these cirumstances.
What would change if the objects were indistinguishable?

I'm interested in a little supporting justification, just so I understand the answer.  Thank you, geniuses!

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