Answer
There are 14 electron states.
Work Step by Step
$n=4, l=3$
In general, the allowed values of $m_l$ are $~~m_l = 0, \pm 1, \pm 2,...,\pm l$
If $l=3,$ then the allowed values of $m_l$ are $~~m_l = 0, \pm 1, \pm 2, \pm 3$
There are 7 possible values of $m_l$
For each value of $m_l$, there are 2 possible values of $m_s$
The number of electron states is $~~(7)(2) = 14$
In general, for a given $l$, the number of electron states is $~~2(2l+1)$
We can verify the number of electron states we found above:
$2(2l+1) = 2[2(3)+1] = 14$
There are 14 electron states.
