Density function for a multivariate Bernoulli-like distribution I'm looking for a distribution to model a vector of $k$ binary random variables, $X_1, \ldots, X_k$.  Suppose I have observed that $\sum_i X_i = n$.  In this case I do not want to treat them as independent Bernoulli random variables.  Instead, I would like something like the multinomial:
$P(X_1=x_1, \ldots, X_k=x_k) = f(x_1, \ldots, x_k; n, p_1, \ldots, p_k) = \frac{n!}{x_1! \cdots x_k!} \prod_{i=1}^k p_i^{x_i}$
but instead of the $x_i$ being nonnegative integers, I want them restricted to be either 0 or 1.  I have been trying to see if the multivariate hypergeometric is appropriate, but I'm not sure.
Thanks in advance for any advice.
 A: The appropriate distribution is Wallenius's noncentral hypergeometric distribution.  Using an urn analogy, the problem is equivalent to picking $n$ of $k$ balls without replacement, where each ball is a different color: the parameters $p$ are analogous to the weights of picking a particular color.
The problem: it's not very convenient to work with, though there is an R package.
A: Update
In light of your comments, here is an updated answer:
Approach 1: Difficult to implement/analyze
Consider the simple case of $k$ = 3 and $n$ = 2. In other words you toss 3 coins (with probabilities $p_1$, $p_2$ and $p_3$). Then, the required mass function for the above case is:
$p_1 p_2 (1-p_3) + p_1 (1-p_2) p_3  + (1-p_1) p_2 p_3$
The above reduces to the binomial if the probabilities $p_i$ are all identical. 
In the general case, you will have ${k \choose n}$ terms where each term is unique with a structure similar to the one above.
Approach 2: Easier to analyze/implement
Instead of the above, you could model each $X_i$ as a bernoulli variable with probability $p_i$. You could then assume that $p_i$ follows a dirichlet distribution.
You would then estimate the model parameters by constructing the posterior distribution for $p_i$ conditional on observing $n$ successes.
If you can normalize by n and and assuming that treating them as probabilities/proportions makes sense in your context you can use the dirichlet distribution.
