  In 1989 Manin and Schechtman defined the discriminantal arrangement
$\mathcal{B}(n, k,\mathcal{A})$ associated to a generic arrangement
$\mathcal{A}$ of $n$ hyperplanes in a $k$-dimensional space. An equivalent
notion was already introduced by Crapo in 1985 with the name of geometry of
circuits. While both those papers were mainly focused on the case in which
$\mathcal{B}(n, k,\mathcal{A})$ has a constant combinatorics when $\mathcal{A}$
changes, it turns out that the case in which the combinatorics of
$\mathcal{B}(n, k,\mathcal{A})$ changes is quite interesting as it classifies
special configurations of points in the $k$-dimensional space. In this paper we
provide an example of this fact elucidating the connection between the well
known generalized Sylvester's and orchard problems and the combinatorics of
$\mathcal{B}(n, k,\mathcal{A})$. In particular we point out how this connection
could be helpful to address those old but still open problems.
