In this basis, the operator $$\hat{P}_+ = \hat{a}^\dagger_1 \hat{a}^\dagger_2 + \hat{a}^\dagger_3 \hat{a}^\dagger_4 + \hat{a}^\dagger_5 \hat{a}^\dagger_6$$ From this we can determine that $$\hat{P}_- | 1, 4, 6 \rangle = \hat{P}_- | 2, 3, 6 \rangle = \hat{P}_- | 2, 4, 5 \rangle = 0$$ so those states all have eigenvalue 0.