 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 $s=1$,Thus the complete list of multiplets isThe total angular momentum operator has squared magnitude eigenvalue ${\hslash }^{2}j(j+1)$ and chosen-axis component eigenvalue $\hslash m_j$. The number of states is $3+5+7=15$, matching the uncoupled dimension $(2\ell +1)(2s+1)=5\times 3$.