Now for further example,
$$
\hat{P}_- | 1,2,3 \rangle = | 3 \rangle
$$
so
$$
\hat{P}_+ \hat{P}_- | 1,2,3\rangle = | 1,2,3\rangle+ | 3,4,3\rangle + | 5,6,3\rangle
$$
The second term vanishes because state 3 is occupied twice, and reordering the last
term we
get
$$
\hat{P}_+ \hat{P}_- | 1,2,3\rangle = | 1,2,3\rangle+ |3, 5,6\rangle
$$
without picking up a phase.