= Solution
For distinguishable particles with one-particle <Hilbert space>[Hilbert spaces] $\mathcal H_1,\ldots,\mathcal H_N$, the <multiparticle quantum state> belongs to
$$
\mathcal H_1\otimes\cdots\otimes\mathcal H_N.
$$
A product of one-particle states gives a product vector, and arbitrary linear combinations give entangled states. A noninteracting Hamiltonian is the sum of one-particle Hamiltonians acting on their respective tensor factors, so product-state energies add.
For two identical particles, the <particle exchange operator> $P_{12}$ swaps every degree of freedom. Since $P_{12}^2=I$, its eigenvalues are $\pm1$. Particles whose allowed total states satisfy
$$
P_{12}|\Psi\rangle=+|\Psi\rangle
$$
are <boson>[bosons], while those satisfying
$$
P_{12}|\Psi\rangle=-|\Psi\rangle
$$
are <fermion>[fermions]. Thus bosonic states are symmetric and fermionic states antisymmetric under interchange. The same condition applies to every transposition in a system of more than two <identical particle>[identical particles].
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