Starting with Eq. (16.4), derive the most probable population distribution for Fermions, namely, Eq. (16.19).

\[\begin{equation*}
W=\prod_{j} \frac{g_{j} !}{\left(g_{j}-N_{j}\right) ! N_{j} !} \tag{16.4}
\end{equation*}\]

\[\begin{equation*}
N_{j}^{*}=\frac{g_{j}}{e^{\alpha} e^{\beta \varepsilon_{j}^{\prime}}+1} \tag{16.19}
\end{equation*}\]