Suppose that $n$ independent trials, each resulting in one of $m$ possible outcomes $1, \ldots , m$ with respective probabilities $p_1, \ldots , p_m$ are performed. Then
$P \{ X_1 = x_1, X_2 = x_2, \ldots, X_m = x_m | \bold{p} \} = \frac{n!}{x_{1}! \cdots x_{m}!} \ p_{1}^{x_{1}} p_{2}^{x_{2}} \cdots p_{m}^{x_{m}}$.
I don't get the following: $\bold{p}$ is chosen by a uniform distribution of the form: $f(p_1, \ldots, p_m) = \begin{cases} c, \ \ 0 \leq p_i \leq 1, i = 1, \ldots m, \sum_{1}^{m} p_i = 1 \\ 0, \ \ \text{otherwise} \end{cases}$ leading to the Bose-Einstein distribution $\binom{n+m-1}{m-1}^{-1}$.