It is known that the carbon-enhanced, extremely metal-poor (CEMP) stars constitute a substantial proportion of the extremely metal-poor (EMP) stars of the Galactic halo, and a by far larger proportion than CH stars among Population II stars. We investigate their origin by taking into account an additional evolutionary path to the surface carbon enrichment, triggered by hydrogen engulfment by the helium flash convection, in EMP stars with [Fe/H]<~-2.5. This process is distinct from the third dredge-up operating in more metal-rich stars and in EMP stars. In binary systems of EMP stars, the secondary stars become CEMP stars through mass transfer from the low- and intermediate- mass primary stars that 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 can reasonably explain the observed properties such as the stellar distributions of the carbon abundances, the binary periods, and the evolutionary stages.