  We introduce the "Median Inverse Problem" for metric spaces. In particular,
having a permutation $\pi$ in the symmetric group $S_n$ (endowed with the
breakpoint distance), we study the set of all $k$-subsets
$\{x_1,...,x_k\}\subset S_n$ for which $\pi$ is a breakpoint median. The set of
all $k$-tuples $(x_1,...,x_k)$ with this property is called the $k$-median
inverse of $\pi$. Finding an upper bound for the cardinality of this set, we
provide an asymptotic upper bound for the probability that $\pi$ is a
breakpoint median of $k$ permutations $\xi_1^{(n)},...,\xi_k^{(n)}$ chosen
uniformly and independently at random from $S_n$.
