Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 33B / b / Solution

For two equal spins, exchange acts on the coupled state $|j,m⟩$ by the Clebsch-Gordan coefficient symmetry factor $(-1{)}^{2j_1-j}$. It is therefore $-1$ when $j=2j_1-1$, making every state in that multiplet antisymmetric, and $+1$ when $j=2j_1-2$, making the next multiplet symmetric.