Mixing and fallback models in faint supernova models are supposed to reproduce the abundance patterns of observed carbon-enhanced metal-poor ( CEMP ) stars in the Galactic halo .
A fine tuning of the model parameters for individual stars is required to reproduce the observed ratios of carbon to iron .
We focus on extremely metal-poor stars formed out of the ejecta from the mixing and fallback models using a chemical evolution model .
Our chemical evolution models take into account the contribution of individual stars to chemical enrichment in host halos together with their evolution in the context of the hierarchical clustering .
Parametrized models of mixing and fallback models for Pop .
III faint supernovae are implemented in the chemical evolution models with merger trees to reproduce the observed CEMP stars .
A variety of choices for model parameters on star formation and metal-pollution by faint supernovae is unable to reproduce the observed stars with [ { Fe } / { H } ] \lesssim - 4 and [ { C } / { H } ] \gtrsim - 2 , which are the majority of CEMP stars among the lowest metallicity stars .
Only possible solution is to form stars from small ejecta mass , which produces an inconsistent metallicity distribution function .
We conclude that not all the CEMP stars are explicable by the mixing and fallback models .
We also tested the contribution of binary mass transfers from AGB stars that are also supposed to reproduce the abundances of known CEMP stars .
This model reasonably reproduces the distribution of carbon and iron abundances simultaneously only if we assume that long-period binaries are favored at [ { Fe } / { H } ] \lesssim - 3.5 .