The production fraction of the ${B}_{c}^{\ensuremath{-}}$ meson with respect to the sum of ${B}^{\ensuremath{-}}$ and ${\overline{B}}^{0}$ mesons is measured in both 7 and 13 TeV center-of-mass (c.m.) energy $pp$ collisions produced by the Large Hadron Collider (LHC), using the LHCb detector. The rate, approximately 3.7 per mille, does not change with energy, but shows a transverse momentum dependence. The ${B}_{c}^{\ensuremath{-}}\ensuremath{-}{B}_{c}^{+}$ production asymmetry is also measured and is consistent with zero within the determined statistical and systematic uncertainties of a few percent.