  Using inelastic scattering of charmed strange mesons by open-charm mesons in
Pb-Pb collisions at the Large Hadron Collider, we study the production of $\psi
(4040)$, $\psi (4160)$, and $\psi (4415)$ mesons. Master rate equations with
the inelastic scattering are established. The scattering is caused by quark
interchange in association with color interactions between all constituent
pairs in different mesons. We consider fifty-one reactions between charmed
strange mesons and open-charm mesons. Unpolarized cross sections for the
reactions are obtained from a temperature-dependent interquark potential.
Temperature dependence of the cross sections leads to that contributions of the
reactions to the production of $\psi (4040)$, $\psi (4160)$, and $\psi (4415)$
change with decreasing temperature during evolution of hadronic matter. For
central Pb-Pb collisions at $\sqrt{s_{NN}}=5.02$ TeV it turned out from the
master rate equations that the $\psi(4040)$ number density is larger than the
$\psi(4160)$ number density which is larger than the $\psi(4415)$ number
density.
