Suppose, we have N distinct urns each having some number of balls $\ge$ 1. The distribution of balls in the urns follows a zero-truncated Poisson distribution with given $\lambda$. If we take out all the balls from all the urns (Say, there are $N_1$ such balls) and randomly select $n_1$ balls from them - it turns out those $n_1$ balls came from $n$ distinct urns (so, $n \le n_1$). So, of course, $n$ is a random variable.
If we know $N, \lambda, n_1$, is it possible to find the distribution of $n$ or the expected value of $n$? I can do it by simulation, but can we find a mathematical expression relating $n$ to $N, \lambda, n_1$?