= Solution
For an intact complex in the usual fast-conformational-averaging regime, counting distinct <chemical shifts> rather than individual split lines gives \b[$C:9$, $D:11$, $E:10$ signals].
In $C$, the uncoordinated pyridinium arm permits the paired positions of a monosubstituted <cyclopentadienyl ligand> to average: its four ring hydrogens give two sets, $2+2$. The linker $\mathrm{CH_2}$ gives one set, the four inequivalent pyridinium C–H positions give four, its N–H gives one, and the two symmetry-related bound <acetonitrile> methyl groups give one. Hence $2+1+4+1+1=9$. The exchangeable N–H resonance may be broadened or absent experimentally; nine is the structural count before exchange removes it.
In the rigid cis-dicarbonyl chelates $D$ and $E$, the different pyridyl-N and fourth-leg donors destroy the operation that related the two sides of the substituted ring. Each of the four Cp hydrogens is distinct, and the two linker hydrogens are <diastereotopic>. Together with four <pyridine> C–H sites, this gives $4+2+4=10$. The remaining bound <acetonitrile> adds one methyl signal to $D$, giving 11; $E$ has none, giving 10. The two <enantiomers> of a racemate have identical spectra in an achiral solvent and do not double these counts. This prediction assumes resolved resonances and no rapid chelate opening that averages inequivalent sites; variable-temperature <proton nuclear magnetic resonance> would test that assumption.
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