Distribution of choice overlap

[Gerenuk]

I have N people and each of them has a uniform probability to belong to one of M countries. Now I wonder what is the distribution of the multiplicities?!
I mean the number of countries with 1 person, 2 persons, 3 persons,... (no matter which country)
Is there an equation for it?

And what if the distribution is not uniform, but I know that some countries are more popular?! (a non-uniform distribution?)
Is there a sensible measure of quantifying this redistribution of popularity?

And does power law distribution play in at some point?

 

[bpet]

And what if the distribution is not uniform, but I know that some countries are more popular?! (a non-uniform distribution?)

If each person has probability p_k of being from country k independently then the distribution of country populations would be multinomial.

The joint distribution of population multiplicities would be a sum of these multinomial probabilities, which for uniform distribution (p_k=1/M) would simplify using the permutations with repeats formula, to get

P(N1=n1,N2=n2,...) = (M!/(n1!n2!...))*(1/M)^N*N!/((1!)^n1.(2!)^n2...)

where N1 is the number of countries with population 1, etc. Not sure how to get the marginal distributions though, or the joint prob for non-uniform.