Notation for Random Bernoulli-Like Vector With Fixed Sum I draw a random vector of dimensionality $k$, each dichotomous element of which taking on a value in $\{0,1\}$. The probability that any element will be $1$ is captured in the $k$-dimensional parameter vector $\bf{p}$, the elements of which are in $[0,1]$ and sum to $1$ (or are normalized as such prior to use). A special condition is that the random vector must sum to some value $c$.
If you're familiar with R, I  can obtain such random vectors in R for some $k$, $c$, and $\bf{p}$ by 
v <- vector(length=k)
v[sample(1:k, c, prob=p)] <- 1
v

I am hoping someone can point me to a distribution that captures this. Ultimately, I need a notation to use to describe this succinctly in a paper.
Thank you very much for your time.
 A: It is immaterial that the $p_i$ sum to unity.  The problem describes a sum of Bernoulli$(p_i)$ distributions and conditions on the sum equalling $c$, $0 \le c \le k$.  (I believe that is as far as we will get in terms of finding names for this procedure.) It is of course a discrete distribution. The nonzero probabilities occur at all vectors having $c$ nonzero components, each potentially with a different probability.  
Vector-like notation is convenient and in common use.  To this end, let us stipulate that


*

*$[k]$ represents the set of indexes $\{1,2,\ldots, k\}$.

*Any vector $v\in \{0,1\}^k$ can be identified with the subset $V\subset [k]$ of indexes $i$ where $v_i \ne 0$.

*The notation $p^V$ means $\prod_{i \in V}p_i = \prod_i p_i^{v_i}$, the product of components of the vector $p$ for which $v_i \ne 0$.

*Let $q_i = 1-p_i$ determine a vector of complementary probabilities $q$.

*Denote by $V^\prime$ the complement of $V$ in $[k]$.

*Write $|V|$ for the cardinality of $V$.
In this notation, the unconditional probability of observing $V$ is $p^Vq^{V^\prime}$.  The probability of observing $V$ for which $|V|=c$ is the sum over all such $V$,
$${\Pr}_p(c) = \sum_{V\subset [k], |V|=c} p^Vq^{V^\prime}.$$
This is a sum over all $\binom{k}{c}$ subsets of size $c$.   There is no simpler formula for it for arbitrary $p$.  Note that $p$ alone determines $k$ and $q$, so nothing else besides $p$ and $c$ needs to be indicated in the notation.
The conditional probability distribution therefore assigns probability
$${\Pr}_{p;c}(V) = \frac{p^Vq^{V^\prime}}{{\Pr}_p(c)}$$
to all $V$ for which $|V|=c$.  In a context in which $p$ is fixed the $p$s can be dropped from the notation; where $c$ is also fixed it may be dropped, too.
