Let’s say $B$ is a discrete set, $B = \{b_1, b_2, \ldots, b_n\}$. If we choose randomly an element of $B$, they are distributed accordingly to $\mathbb{P}( \text{choose }b_i) = p_i$. More formally, let $\tilde{b}$ be a random variable that picks one $b_i$ at random, then $\mathbb{P}(\tilde{b} = b_i) = p_i$.
Now, let $X$ be the random variable that selects a subset of $B$ of $k$ elements according to the distribution of $\tilde{b}$. My question is, given a subset containing $k$ elements, $\bar{B} = \{b_{i_1}, \ldots, b_{i_k}\}$, is it possible to explicitely compute the probability distribution of $X$, i.e., $ \mathbb{P}(X = \bar{B}) $?