= Solution
For <identical particles>, the <particle exchange operator> exchanges both spatial and <spin> labels. The <symmetrization postulate> requires a symmetric full state for <bosons> and an antisymmetric full state for <fermions>. A product $\Psi=\phi\chi$ must therefore have spatial and spin exchange eigenvalues whose product is $+1$ or $-1$, respectively; a general state may be a sum of such products.
For orthonormal spatial orbitals $u,v$, the combinations $\phi_\pm=[u(1)v(2)\pm v(1)u(2)]/\sqrt2$ have opposite exchange parities. Two spin-one-half <fermions> combine $\phi_+$ with the antisymmetric <spin-one-half singlet state> $(|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle)/\sqrt2$, or $\phi_-$ with one of the three symmetric <spin-one-half triplet states>. \b[Two particles in the same spatial orbital must be a spin singlet.] This is the <Pauli exclusion principle> applied to complete <spin orbitals>, not a ban on sharing a spatial orbital with opposite <spins>.
A <Slater determinant> enforces fermionic antisymmetry for many occupied <spin orbitals>. Bosonic symmetrization instead uses a <permanent>; repeated orbital occupations are allowed. These exchange requirements apply even when particles do not interact.
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