= Solution
{c}
For two decays from a common upper <energy level>, their photon <wavenumber> difference equals the separation of the two lower <energy levels>. Treating both quoted <wavelengths> as vacuum values initially gives
$$
\widetilde\nu_1=\frac{10^7}{193.09}=51789.32\,\mathrm{cm^{-1}},\qquad
\widetilde\nu_2=\frac{10^7}{247.86}=40345.36\,\mathrm{cm^{-1}},
$$
whose difference $11443.96\,\mathrm{cm^{-1}}$ is close to $21648-10192=11456\,\mathrm{cm^{-1}}$. The only matching lower pair among the supplied <energy levels> is ${}^1D_2,{}^1S_0$, with the shorter-<wavelength> <photon> ending on ${}^1D_2$. The two independent upper <energy> estimates are
$$
\widetilde E_u=10192+51789.32=61981.32\,\mathrm{cm^{-1}},\qquad
\widetilde E_u=21648+40345.36=61993.36\,\mathrm{cm^{-1}}.
$$
Thus
$$
\boxed{\widetilde E_u\simeq6.20\times10^4\,\mathrm{cm^{-1}}\simeq7.69\,\mathrm{eV},\quad
1s^22s^22p3s\,{}^1P_1^{\rm o}.}
$$
The <electric-dipole selection rules> require odd <parity>, $S=0$, $L=1$ and $J=1$ to connect to both singlet lower <energy levels>. The apparent discrepancy has a concrete convention explanation. The NIST <carbon> line table lists $193.0906\,\mathrm{nm}$ under \b[vacuum] <wavelengths> and $247.8561\,\mathrm{nm}$ under \b[air] <wavelengths>, reproducing the source's rounded numbers.<Strong Lines of Carbon by National Institute of Standards and Technology>[Vacuum section: C I 1930.906 Å; air section: C I 2478.561 Å]{ref} The second number must therefore be converted to a vacuum <wavelength> before using $E=hc/\lambda$. For the tabulated upper $61981.82\,\mathrm{cm^{-1}}$ and lower $21648.02\,\mathrm{cm^{-1}}$ <energy levels>, its vacuum <wavelength> is $247.931\,\mathrm{nm}$, corresponding to an air <refractive index> about $1.00030$. This removes the spurious $12\,\mathrm{cm^{-1}}$ discrepancy. The first calculation alone, which correctly uses a vacuum <wavelength>, already gives the intended upper <energy level> to the accuracy of the rounded data. The independent upper value is $61981.82\,\mathrm{cm^{-1}}$.<Energy Levels of Neutral Carbon by National Institute of Standards and Technology>[2s²2p(²P°)3s ¹P°₁ entry at 61981.82 cm⁻¹]{ref}
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