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If both electrons are in ground state, characterized by $(1s)^2$, this means that n =1, l = 0. The two electrons occupy the same energy and space (why doesn't pauli exclusion principle prevent this?), so the spatial part is symmetric.

So in order for the state to obey fermi-dirac statistics, the overall wavefunction (spatial*spin) must be anti-symmetric.

Does this mean that the spin part is necessarily anti-symmetric? Thus it must be a tripplet state.

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The Pauli exclusion principle does not prevent it precisely because the spin is antisymmetric, as you said. The anti-symmetric spin state, however, is a singlet. This is because there is only one way to form an anti-symmetric two-particle state from only two possible one-particle states, as explained in this answer.

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