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.