  Following the proof of the purity conjecture for weakly separated
collections, recent years have revealed a variety of wider examples of purity
in different settings. In this paper we consider the collection $\mathcal
A_{I,J}$ of sets that are weakly separated from two fixed sets $I$ and $J$. We
show that all maximal by inclusion weakly separated collections $\mathcal
W\subset\mathcal A_{I,J}$ are also maximal by size, provided that $I$ and $J$
are sufficiently "generic". We also give a simple formula for the cardinality
of $\mathcal W$ in terms of $I$ and $J$. We apply our result to calculate the
cluster distance and to give lower bounds on the mutation distance between
cluster variables in the cluster algebra structure on the coordinate ring of
the Grassmannian. Using a linear projection that relates weak separation to the
octahedron recurrence, we also find the exact mutation distances and cluster
distances for a family of cluster variables.
