  CEMP-$r/s$ stars are metal-poor stars with enhanced abundances of carbon and
heavy elements associated with the slow ($s$-) and rapid ($r$-) neutron-capture
process. It is believed that carbon and $s$-elements were accreted from the
wind of an AGB primary star, a scenario that is generally accepted to explain
the formation of CEMP stars that are only enhanced in $s$-elements (CEMP-$s$
stars). The origin of $r$-element-enrichment in CEMP-$r/s$ stars is debated and
many formation scenarios have been put forward. We aim to determine the
likelihood of the scenarios proposed to explain the formation of CEMP-$r/s$
stars. We calculate the frequency of CEMP-$r/s$ stars among CEMP-$s$ stars for
a variety of scenarios, and we compare it with that determined from an observed
sample of CEMP-$r/s$ stars collected from the literature. The theoretical
frequency of CEMP-$r/s$ stars predicted in most scenarios underestimates the
observed ratio by at least a factor of 5. If the enrichments in $s$- and
$r$-elements are independent, the model ratio of CEMP-$r/s$ to CEMP-$s$ stars
is about 22%, that is approximately consistent with the lowest estimate of the
observed ratio. However, this model predicts that about one third of all
carbon-normal stars have [Ba/Fe] and [Eu/Fe] higher than 1, and that 40% of all
CEMP stars have [Ba/Eu]$\le0$. Stars with these properties are at least ten
times rarer in our observed sample. The $intermediate$ or $i$-process, which is
supposedly active in some circumstances during the AGB phase, could provide an
explanation of the origin of CEMP-$r/s$ stars, similar to that of CEMP-$s$
stars, in the context of wind mass accretion in binary systems. Further
calculations of the nucleosynthesis of the $i$-process and of the detailed
evolution of late AGB stars are needed to investigate if this scenario predicts
a CEMP-$r/s$ star frequency consistent with the observations.
