= Solution
Take the eight Mo–Cl bond-length changes as a stretching basis. These scalar coordinates have the same permutation representation as the eight sigma donors:
$$
\Gamma_{\mathrm{stretch}}=A_{1g}+B_{1g}+E_g+A_{2u}+B_{2u}+E_u.
$$
<Infrared spectroscopy> detects modes transforming as a dipole component: $z$ is $A_{2u}$ and $(x,y)$ is $E_u$. There are therefore \b[two IR fundamentals], one $A_{2u}$ and one doubly degenerate $E_u$ frequency.
<Raman spectroscopy> detects modes transforming as polarizability components. The relevant modes are $A_{1g}$, $B_{1g}$ and $E_g$, giving \b[three Raman fundamentals], of which $E_g$ is doubly degenerate. The remaining $B_{2u}$ stretch is silent in both ordinary spectra.
Thus the prediction is
$$
\boxed{N_{\mathrm{IR}}=2,\qquad N_{\mathrm{Raman}}=3,\qquad
N_{\mathrm{coincident\ modes}}=0.}
$$
The zero follows from <infrared and Raman mutual exclusion>: the molecule has inversion symmetry, dipole components are ungerade, and polarizability components gerade. This counts resolved fundamental frequencies of the ideal ion, not separate lines for degenerate components, isotope splittings or accidental numerical overlaps between distinct modes.
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