The CHemical Abundances Of Spirals (CHAOS) project is building a large database of Large Binocular Telescope (LBT) H ii region spectra in nearby spiral galaxies to use direct abundances to better determine the dispersion in metallicity as a function of galactic radius. Here, we present CHAOS LBT observations of C ii λ4267 emission detected in 10 H ii regions in M 101, and using a new photoionization model-based ionization correction factor, we convert these measurements into total carbon abundances. A comparison with M 101 C ii recombination line observations from the literature shows excellent agreement, and we measure a relatively steep gradient in log(C/H) of-0.37 ± 0.06 dex R-1 e. The C/N observations are consistent with a constant value of log(C/N) = 0.84 with a dispersion of only 0.09 dex, which, given the different nucleosynthetic sources of C and N, is challenging to understand. We also note that when plotting N/O versus O/H, all of the H ii regions with detections of C ii λ4267 present N/O abundances at the minimum of the scatter in N/O at a given value of O/H. If the high surface brightness necessary for the detection of the faint recombination lines is interpreted as an indicator of H ii region youth, then this may point to a lack of nitrogen pollution in the youngest H ii regions. In the future, we anticipate that the CHAOS project will significantly increase the total number of C ii λ4267 measurements in extragalactic H ii regions.