Answer
See the detailed answer below.
Work Step by Step
$$\color{blue}{\bf [a]}$$
The transition of $4p \to 4s$ is allowed because it satisfies the selection rule where the change in the orbital angular momentum quantum number ($ \Delta l $) is equal to 1. Using the sodium energy levels, the wavelength can be calculated using the formula of
$$
\lambda = \frac{hc}{\Delta E}
$$
where the energy difference between the $4p$ and $4s$ levels $ \Delta E $ is $ 3.75 \, \text{eV} - 3.19 \, \text{eV} $
$$
\lambda = \frac{(6.63\times 10^{-34})(3\times 10^8)}{(3.75 - 3.19)(1.6\times 10^{-19}) } = \color{red}{\bf 2220} \, \text{nm}
$$
$$\color{blue}{\bf [b]}$$
The transition of $3d \to 4s$ is not allowed because it violates the selection rule that requires the change in angular momentum quantum number ($ \Delta l $) to be 1. In this case, the change would be $ \Delta l = 2 $, making the transition forbidden.
