To match List I (Quantum Numbers) with List II (Information provided), we need to understand what each quantum number represents:

1. Principal Quantum Number ($n$): This quantum number determines the size and energy level of the orbital. An increase in $n$ implies a higher energy level and a larger orbital size.

2. Azimuthal Quantum Number ($\ell$): (not directly listed but related to $\mathrm{m_l}$ because $\mathrm{m_l}$ depends on $\ell$): This quantum number defines the shape of the orbital. Different values of $\ell$ correspond to different shapes (s, p, d, f, etc.).

3. Magnetic Quantum Number ($\mathrm{m_l}$): This quantum number describes the orientation of the orbital in space. It can take values from $-\ell$ to $\ell$, where each value corresponds to a specific orientation of the orbital.

4. Spin Quantum Number ($\mathrm{m_s}$): This quantum number specifies the orientation of the spin of the electron. The two possible values are $+\frac{1}{2}$ and $-\frac{1}{2}$, representing the two possible spin orientations (up or down).

Matching the given options:

A. $\mathrm{m_l}$ - should be matched with "Orientation of orbital" (III).

B. $\mathrm{m_s}$ - should be matched with "Orientation of spin of electron" (IV).

C. I - This is likely intended to be $\ell$, and would be matched with "Shape of orbital" (I).

D. $n$ - relates to "Size of orbital" (II), as it affects the distance from the nucleus along with the energy level.

Using the understanding above, we can identify the correct matches:

A-III (Orientation of orbital)

B-IV (Orientation of spin of electron)

C-I (Shape of orbital)

D-II (Size of orbital)

Therefore, the correct answer is:

Option B: A-III, B-IV, C-I, D-II