= Solution
For <addition of angular momentum>, the allowed total <quantum number> runs in unit steps from $|\ell-s|$ to $\ell+s$. With $\ell=2$ and $s=1$,
$$
\boxed{j=1,2,3,\qquad m_j=-j,-j+1,\ldots,j.}
$$
Thus the complete list of multiplets is
$$
\begin{array}{c|l}
j&m_j\\\hline
1&-1,0,1\\
2&-2,-1,0,1,2\\
3&-3,-2,-1,0,1,2,3
\end{array}
$$
The <total angular momentum operator> has squared magnitude <eigenvalue> $\hbar^2j(j+1)$ and chosen-axis component <eigenvalue> $\hbar m_j$. The number of states is $3+5+7=15$, matching the uncoupled dimension $(2\ell+1)(2s+1)=5\times3$.
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