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.
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