Suppose that $\displaystyle n $ independent trials, each resulting in one of $\displaystyle m $ possible outcomes $\displaystyle 1, \ldots , m $ with respective probabilities $\displaystyle p_1, \ldots , p_m $ are performed. Then

$\displaystyle 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: $\displaystyle \bold{p} $ is chosen by a uniform distribution of the form: $\displaystyle 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 $\displaystyle \binom{n+m-1}{m-1}^{-1} $.