Proof
Enumerate all of the n data items individually with the \(n_1!\) identical values first and the remaining groups in like manner to get the enumerated list
\begin{equation*}
\left \{ x_{1,1}, ..., x_{1,n_1}, x_{2,1}, ..., x_{2,n_2}, ... , x_{s,1}, ..., x_{s,n_s} \right \}
\end{equation*}
In this order, there are \(n_1!\) ways to arrange the first group, \(n_2!\) ways to arrange the second, etc. There are \(n_1! \times n_2! \times ... \times \n_s!\) ways to arrange all of the categories together with groups in this order but none of those group reorders does anything since those data values are all the same. Dividing out those from the \(n!\) original permutations of all items leaves one with the multinomial coefficient.