Answer
See work below.
Work Step by Step
a. The values of $m_{\mathcal{l}}$ range from $-\mathcal{l}$ to $\mathcal{l}$, or $2\mathcal{l}+1$ values. For each of those values, there are 2 possible values of $m_{s}$, which are $-\frac{1}{2},\frac{1}{2}$.
We see that the total number of states for a particular $\mathcal{l}$ is $2(2\mathcal{l}+1)$.
b. For $\mathcal{l}$ = 0, 1, 2, 3, 4, 5, and 6, the number of states is 2, 6, 10, 14, 18, 22, and 26, respectively.
