\begin{equation} \label{eq:prob_us}
    \begin{split}
        P &= \sum_{i = 1} \sum_{j = 1} \binom{i + j}{i} \binom{n}{i + j} \\ 
        & \times \frac{N_{1}! \, N_{2}! \, N_{out}! \left(N - n\right)!}{\left(N_{1} - i\right)! \, \left(N_{2} - j\right)! \, \left(N_{out} - \left(n - \left(i + j\right)\right)\right)! \, N!},
    \end{split}
\end{equation}
