The CHAOS project is building a large database of 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 $lambda$4267 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 M101 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_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 CII $lambda$4267 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 $lambda$4267 measurements in extragalactic H II regions.
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