The spherical harmonics form an orthonormal basis, and the given state is normalized, so we can add up the probabilities that the the molecule is observed in each of the spherical harmonic states with total angular momentum $5$ i.e. in each of the $Y_5^m$ states. The probability to measure $l$ and $m$ for the total angular momentum and the azimuthal angular momentum quantum number is $\left| \langle Y_l^m | \Psi \rangle \right|^2$ , so the desired probability is:

$\begin{align*}\sum_{m = -5}^{+5} \left| \langle Y_l^m | \Psi \rangle \right|^2 = \frac{3^2}{38} + \frac{2^2}{38} = \boxed{\f...$
Therefore, answer (C) is correct.

Solution to 1996 Problem 33