We have mapped the entire TMC-1C region in the (HCO+)-C-13 (J = 1-0) and CH3OH (J(K) = 2(0)-1(0)A(+)) lines at a grid spacing of 50" with the 45 m telescope at Nobeyama Radio Observatory. We have also conducted high spatial resolution mapping observations of the TMC-1C region at a grid spacing of 34 " in both lines toward a 6' x 6' portion in the south and a 4' x 4' portion in the north. We found that the structure of TMC-1C is filamentary in both molecular lines. The size and position angle of the filament are 0.75 pc x 0.17 pc and 135 degrees, respectively. The filament consists of dense (similar to 10(5) cm(-3)) cores which are traced by either (HCO+)-C-13 or CH3OH lines. We found that the distribution of cores seen in (HCO+)-C-13 is quite different from the distribution of cores seen in CH3OH. The large velocity gradient analyses indicate that this difference is a result of the relative abundance variation between (HCO+)-C-13 and CH3OH in the cores by about 1 order of magnitude. We have also carried out multitransitional observations of C3H2 (J(K',K ") = 2(1,2)-1(0,1) and 3(1,2)-3(0,3)) at two positions in the same cloud in order to estimate the molecular hydrogen densities for H13CO+ and CH3OH cores and found that the densities are around 10(5) cm(-3) for both cores. These starless cores (no IRAS source), considered to be prestellar cores, seem to be at chemically different evolutionary stages; the (HCO+)-C-13 cores are more evolved and closer to protostar formation than CH3OH cores. On the other hand, we found no difference in physical properties, i.e., the size, line width, and mass, between (HCO+)-C-13 and CH3OH cores; the averages are about 0.07 pc, 0.3 km s(-1), and 2 M-., respectively.