Comparative Study of the Hadronic Production of $B_c$ Mesons

The full order $\alpha_s^4$ perturbative QCD calculation of the production of $B_c$ mesons at the Tevatron is compared with the fragmentation approximation. The non-fragmentation diagrams, in which two or more quarks and/or gluons can simultaneously be nearly on-shell, are important unless $P_T\gg M_{B_c}$.