The number of trials N, \mathbf{x} = (x_v)_{v=1}^V, x_v represents the number of v occuring, \phi_v represents the probability of v occuring.
\begin{align*}
\mathrm{Mult}(x|N, \set{\phi_v}_{v=1}^V) = \frac{N!}{x_1! x_2! \dots x_V!} \prod_{v=1}^V \phi_v^{x_v}
\end{align*}
\begin{align*}
E[x_v] &= N \phi_v \\
V[x_v] &= N \phi_v (1 - \phi_v)
\end{align*}
Categorical distributions are defined as follows:
\begin{align*}
\mathrm{Cat}(x|\set{\phi_v}_{v=1}^V) = \phi_x ,\quad x \in \set{1, 2, \dots, V}
\end{align*}
The relationships between multinomial distributions and categorical and binomial distributions are as follows:
\begin{align*}
\mathrm{Mult}(x|1, \set{\phi_v}_{v=1}^V) &= \mathrm{Cat}(x|\set{\phi_v}_{v=1}^V) \\
\mathrm{Mult}(x|N, \set{\phi_v}_{v=1}^2) &= \mathrm{Bin}(x|N, \phi_1) \\
\end{align*}
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