Counting the States for $\vert\ell_1-\ell_2\vert\leq j\leq\ell_1+\ell_2$.

If we add $\ell_1$ to $\ell_2$ there are $(2\ell_1+1)(2\ell_2+1)$ product states. Lets add up the number of states of total $\ell$. To keep things simple we assume we ordered things so $\ell_1\geq\ell_2$.

$$\bgroup\color{black}\sum\limits_{\ell=\ell_1-\ell_2}^{\ell_1+... ...l_2+1)(2\ell_1-2\ell_2+1)+2\sum\limits_{n=0}^{2\ell_2}n \egroup$$

$$\bgroup\color{black}=(2\ell_2+1)(2\ell_1-2\ell_2+1)+(2\ell_2+1)(2\ell_2) =(2\ell_2+1)(2\ell_1+1)\egroup$$

This is what we expect.