Problem
The multinominal Dirichlet distribution is defined for probability vector $\mathbf{p}=[p_1,\cdots, p_d]^T$ with parameter $\alpha=[\alpha_1,\cdots,\alpha_d]^T$ $$ \pi(\mathbf{p})=\frac{\Gamma(|\alpha|)}{\prod_{j=1}^{d} \Gamma\left(\alpha_{j}\right)} \prod_{j=1}^{d} p_{j}^{\alpha_{j}-1} $$ where $\vert\alpha\vert=\sum_{i=1}^d\alpha_i$
Now for vector $\mathbf{x}=[x_1,\cdots,x_n]^T$ where each $x_i \in \{0,1\}$ and $\sum_{i=1}^n x_i=16$, find its PMF under multinominal Dirichlet distribution.
What I Have Done
It is not difficult to write a general form formula $$ f(\mathbf{x} | \alpha)=\int_{\Delta_{d}} \left( \begin{array}{c}{\sum_{i=1}^dx_i} \\ {\mathbf{x}}\end{array}\right) \prod_{j=1}^{d} p_{j}^{x_{j}} \pi(\mathbf{p}) d \mathbf{p} $$ where $\Delta d$ is a unit simplex.
However, I am doubtful if this could actually give a closed-form formula. More specifically, I am not sure how to deal with vector $\mathbf{p}$ and product in the integrand.
Solution
The PMF result is available on Wikipedia. For those who interested in derivation, a complete derivation could be found here.