Counting states for Arbitrary $\ell$ Plus spin ${1\over 2}$

For angular momentum quantum number $\ell$, there are $(2\ell + 1)$ different m states, while for spin we have 2 states $\chi_{\pm}$. Hence the composite system has $2(2\ell + 1)$ states total.

Max $j_z = \ell + {1\over 2}$ so we have a state with $j=\ell+{1\over 2}$. This makes up $(2j+1)=(2\ell+2)$ states, leaving

$$\bgroup\color{black}(4\ell + 2) - (2\ell + 2) = 2\ell = \left(2\left(\ell-{1\over 2}\right) +1\right)\egroup$$

Thus we have a state with $j=\ell-{1\over 2}$ and that's all.