  In this paper, the implication of the extreme large $N_c$ limit for
nucleon-nucleon cross section is examined. Starting from the $N_c$ scaling of
the S-matrix elements, a relation between the total inelastic and total cross
sections is derived for the regime in which the momenta are much larger than
the QCD scale. A conceptual complication arise from the fact that there is a
tower of baryons, such as the $\Delta$, with mass splittings from the nucleon
which go to zero at large $N_c$. Since these baryons are stable at large $N_c$,
the meaning of elasticity must be modified. Processes which only transform
nucleons to these baryons are considered elastic; processes in which at least
one additional meson is produced are considered inelastic. It is shown that in
the extreme large $N_c$ limit, the total inelastic cross section is exactly one
eighth of the total cross section. In contrast, in the physical world of
$N_c=3$ the total cross section for high-energy nucleon-nucleon scattering is
dominated by inelastic processes.
