We present new theoretical stellar evolutionary models of metal-rich asymptotic giant branch (AGB) stars. Stellar models are evolved with initial masses between 1 and 7 M⊙ at Z = 0.007, and 1 and 8 M⊙ at Z = 0.014 (solar) and at Z = 0.03. We evolve models with a canonical helium abundance and with helium-enriched compositions (Y = 0.30, 0.35, and 0.40) at Z = 0.014 and 0.03. The efficiency of third dredge-up and the mass range of carbon stars decreases with an increase in metallicity. We predict carbon stars form from initial masses between 1.75 and 7 M⊙ at Z = 0.007 and between 2 and 4.5 M⊙ at solar metallicity. At Z = 0.03, the mass range for C-star production is narrowed to 3.25–4 M⊙. The third dredge-up is reduced when the helium content of the model increases owing to the reduced number of thermal pulses on the AGB. A small increase of ΔY = 0.05 is enough to prevent the formation of C stars at Z = 0.03, depending on the mass-loss rate, whereas at Z = 0.014, an increase of ΔY ≳ 0.1 is required to prevent the formation of C stars. We speculate that the probability of finding C stars in a stellar population depends as much on the helium abundance as on the metallicity. To explain the paucity of C stars in the inner region of M31, we conclude that the observed stars have Y ≳ 0.35 or that the stellar metallicity is higher than [Fe/H] ≈ 0.1.