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It's because first homotopy group for configuration space of one-particle states in 3D space is permutations group. In two words: scalar product $\langle \mathbf p_{1}, \mathbf p_{2}, ...| \mathbf k_{1}, \mathbf k_{2} , ...\rangle$ is invariant under simultaneous permutations $\mathbf p$ and $\mathbf k$ for different particles. For identical particles, ...
Sometimes when people try to simplify physics they go too far and oversimplify. An example would be if people tell you that two atoms of $He^4$ can be in the same state because they "are bosons." First let's look at one fermion, say an electron. It has a spin and a spatial extent, the general state of the electron is a product of the spin and the spatial ...