  The cross-section for the production of $b{\bar b} c{\bar c}$ quarks in
$e^+e^-$ annihilation, that proves to be at a level of $\sigma (e^+e^-
\rightarrow b{\bar b} c{\bar c})/
  \sigma (e^+e^- \rightarrow b{\bar b}) \sim 10^{-2}$ for ${\sqrt s}=M_Z$ is
calculated within the frames of the QCD perturbation theory. The cross sections
for the associated production of $1S$- and $2S$-wave states of $B_c$-meson in
the reaction $e^+e^- \rightarrow B_c b{\bar c}$ were calculated in the
nonrelativistic model of a heavy quarkonium. The fragmentation function of
$b\to B_c^{(*)}$ is analysed in the scaling limit. The number of
$\Lambda_{bc}$-hyperons to be expected at LEP is estimated on the basis of the
assumption on quark-hadron duality.
