Answer
a. 16.
b. -0.85 eV.
Work Step by Step
a. For n = 4, $l$ can take on the values 3, 2, 1, and 0.
For each $l$ value, $m_l$ can be $\pm l, \pm (l-1)…0$.
Here are the possible $(l, m_l)$ pairs for n =4:
$$(l=3, m_l=3) (l=3, m_l=2) (l=3, m_l=1) (l=3, m_l=0) (l=3, m_l=-1) (l=3, m_l=-2) (l=3, m_l=-3)$$
$$(l=2, m_l=2) (l=2, m_l=1) (l=2, m_l=0) (l=2, m_l=-1) (l=2, m_l=-2) $$
$$(l=1, m_l=1) (l=1, m_l=0) (l=1, m_l=-1) $$
$$(l=0, m_l=0) $$
There are $n^2=16$ combinations.
b. Each of the 16 states has the same energy because the energy depends only on n, which is 4 for all of them.
$$E=-\frac{13.6eV}{4^2}=-0.85eV$$
