  The stellar population in the Galactic halo is characterised by a large
fraction of CEMP stars. Most CEMP stars are enriched in $s$-elements (CEMP-$s$
stars), and some of these are also enriched in $r$-elements (CEMP-$s/r$ stars).
One formation scenario proposed for CEMP stars invokes wind mass transfer in
the past from a TP-AGB primary star to a less massive companion star which is
presently observed. We generate low-metallicity populations of binary stars to
reproduce the observed CEMP-star fraction. In addition, we aim to constrain our
wind mass-transfer model and investigate under which conditions our synthetic
populations reproduce observed abundance distributions. We compare the CEMP
fractions and the abundance distributions determined from our synthetic
populations with observations. Several physical parameters of the binary
stellar population of the halo are uncertain, e.g. the initial mass function,
the mass-ratio and orbital-period distributions, and the binary fraction. We
vary the assumptions in our model about these parameters, as well as the wind
mass-transfer process, and study the consequent variations of our synthetic
CEMP population. The CEMP fractions calculated in our synthetic populations
vary between 7% and 17%, a range consistent with the CEMP fractions among very
metal-poor stars recently derived from the SDSS/SEGUE data sample. The results
of our comparison between the modelled and observed abundance distributions are
different for CEMP-$s/r$ stars and for CEMP-$s$ stars. For the latter, our
simulations qualitatively reproduce the observed distributions of C, Na, Sr,
Ba, Eu, and Pb. Contrarily, for CEMP-$s/r$ stars our model cannot reproduce the
large abundances of neutron-rich elements such as Ba, Eu, and Pb. This result
is consistent with previous studies, and suggests that CEMP-$s/r$ stars
experienced a different nucleosynthesis history to CEMP-$s$ stars.
