  We present new Hubble Space Telescope WFC3/IR medium-band photometry of the
compact elliptical galaxy M32, chemically resolving its thermally pulsating
asymptotic giant branch stars. We find 2829 M-type stars and 57 C stars. The
carbon stars are likely contaminants from M31. If carbon stars are present in
M32 they are so in very low numbers. The uncorrected C/M ratio is 0.020 $\pm$
0.003; this drops to less than 0.007 after taking into account contamination
from M31. As the mean metallicity of M32 is just below solar, this low ratio of
C to M stars is unlikely due to a metallicity ceiling for the formation of
carbon stars. Instead, the age of the AGB population is likely to be the
primary factor. The ratio of AGB to RGB stars in M32 is similar to that of the
inner disc of M31 which contain stars that formed 1.5-4 Gyr ago. If the M32
population is at the older end of this age then its lack of C-stars may be
consistent with a narrow mass range for carbon star formation predicted by some
stellar evolution models. Applying our chemical classifications to the dusty
variable stars identified with {\em Spitzer}, we find that the x-AGB candidates
identified with Spitzer are predominately M-type stars. This substantially
increases the lower limit to the cumulative dust-production rate in M32 to $>$
1.20 $\times 10^{-5}$ ${\rm M}_{\odot} \, {\rm yr}^{-1}$.
