  It is known that the carbon-enhanced, extremely metal-poor (CEMP) stars
constitute a substantial proportion in the extremely metal-poor (EMP) stars of
the Galactic Halo, by far larger than CH stars in Population II stars. We
investigate their origin with taking into account an additional evolutionary
path to the surface carbon-enrichment, triggered by hydrogen engulfment by the
helium flash convection, in EMP stars of $[Fe/H] \lesssim -2.5$. This process
is distinct from the third dredge-up operating in more metal-rich stars and
also in EMP stars. In binary systems of EMP stars, the secondary stars become
CEMP stars through mass transfer from the primary stars of low and intermediate
masses, which have developed the surface carbon-enhancement. Our binary
scenario can predict the variations in the abundances not only for carbon but
also for nitrogen and s-process elements and reasonably explain the observed
properties such as the stellar distributions with respect to the carbon
abundances, the binary periods, and the evolutionary stages. Furthermore, from
the observed frequencies of CEMP stars with and without s-process element
enhancement, we demonstrate that the initial mass function of EMP stars need to
give the mean mass $~10\msun$ under the reasonable assumptions on the
distributions of orbital separations and mass ratio of binary components. This
also indicates that the currently observed EMP stars were exclusively born as
the secondary members of binaries, making up $\sim 10%$ remnants of EMP binary
systems of mass $~10^8\msun$ in total; in addition to CEMP stars with white
dwarf companions, a significant fraction of them have experienced supernova
explosions of their companions. We discuss the implications of the present
results in relation to the formation of Galactic halo.
