  Using the Thomas-Fermi quark model, a collective, spherically symmetric
density of states is created to represent a gas of interacting fermions with
various degeneracies at zero temperature. Over a family of multi-pentaquarks,
color interaction probabilities are obtained after averaging over all the
possible configurations. It is found that three different Thomas-Fermi
functions are necessary for light, charm, and anti-charm quarks. These are
assumed to be linearly related by proportionality constants resulting in
consistency conditions. We analyze these conditions and find that they lead to
an interesting pattern of spherical quark distributions.
