The input to the problem is a set of "addresses" represented by integers:

example:
input = [14 , 89 , 32 , 1231 , 92310 , 3985 , 291 ... ]

Each address belongs to a server.  A single server maps to multiple
addresses.  Thus, within each input set, there are multiple subsets of
addresses that belong to the same server:

example:
input = [ [14, 874, 32] , [99912, 1232, 245, 23325] , ... [ ... ] ]

The input can be described by the size of its subsets as follows:

[ 3 , 4 , 10, 2 ...] 

Where each number represents the number of distinct addresses in the
input belonging to a single server.

The question is:

If you choose C addresses from the input, what is the probability of
finding at least one subset of size greater than or equal to G that
belong to the same server within those C addresses?

I am looking for at least a poly-time algorithm that does not involve
calculating the solution recurisively or with a brute force.

My impression is that, given the high variability of input data, a
recursive formula would be the efficient way to solve such problems.

If we specify both a partition of N, together with integers C no more
than N and G no more than C, and ask what is the probability of
choosing C of N "addresses" such that at least G of them belong to the
same subset of the partition, then a recursive formula can be given. 
Whether such an approach is satisfactory is of course a matter about
which the Customer is the expert, but it appears to me to offer a
practical way of obtaining the result, using a computer program, for
moderate values of G and C and a fixed partition of N.

In particular each step of the recursion, while involving a number of
terms, would reduce the number of parts used in a partition by at
least one (keeping G as high as its original setting).  I'd have to do
more implementation/analysis to be sure, but it looks like the basis
for a practical method.

regards, mathtalk-ga