I have a set of $N(\geq 2)$ objects which I randomly group in $C(\leq N)$ clusters i.e. all the $C$ clusters have atleast one object and all such clusterings are equally likely.
What is the probability that 2 particular objects from this group of $N$ objects are in same cluster?
I don't know if it is what you ask, but the answer with fixed clusters can be derived in this way:
What is the probability that 2 particular objects belong to the kth cluster?
The probability is the number of manners to combine two individuals from cluster $k$ ($C_k$ is the number of objects on the cluster $k$) divided by the number of manners to combine two individuals from $N$ of them (the total number of pairs).
$(_{2}^{C_k})/(_{2}^{N})$.
What is the probability that the 2 particular objects belong to the same cluster? Since the pair belongs to the cluster 1 or cluster 2 or ... or cluster C, we have:
It is the probability that the 2 objects belong to the cluster $1$ (with $C_1$ objects) plus the probability that the 2 objects belong to the cluster 2 (with $C_2$ objects) plus .... plus the probability that the 2 objects belong to in the cluster C (with $C_c$ objects)
$(_{2}^{N})^{-1} \sum_{k=1}^{C} (_{2}^{C_k})$,
where $C$ is the number of clusters.
Edit any error, please!