Suppose a sample $X_1, X_2, ...$ follows a Binomial distribution with parameters $m \in \mathbb N$ and $\pi \in (0, 1)$. Denote \begin{equation} Z_k := \sum_i I(X_i = k) \quad \text{for} \quad k = 0, 1, \ldots, m, \end{equation}
where $I(\cdot)$ is the indicator function.
This particular problem concerns the case of missing data for $Z_0$. We are given that $\sum_{k=1}^m Z_k =: n$. Moreover, $Z_0 + n$ follows a negative binomial distribution with parameters $r = n$ and \begin{equation} \theta = \mathbb P (X_i \geqslant 1) = 1 - (1 - p)^m. \end{equation}
The question is, what is the joint distribution of $Z_0, \ldots, Z_m$, i.e. what is \begin{equation} \mathbb P (Z_0 = z_0, \ldots, Z_6 = z_6). \end{equation}
Normally I would say the distribution is Multinomial with class probabilities \begin{equation} \pi_k = \mathbb P (X_i = k) \quad \text{for} \quad k = 0, \ldots, m, \end{equation}
but the size parameter cannot be $n + Z_0$. Any suggestions?
Thanks in advance!