Antisymmetry of the helium wavefunction This question is about an assignment I was given at my q.m. course at uni.
The task is to list all possible states (in spectroscopic notation) for the He configuration $ns^1 n's^1$.
The solutions are given as $^1S$ and $^3S$.
How can both of these be possible without breaking the Pauli principle?
My understanding is that the total wave function (consisting of both spin and position-space wave function) has to be antisymmetric.
For $^1S$, the spin wave function is antisymmetric, meaning the position-space wave function has to be symmetrical, while for $^3S$ the opposite would have to be true.
Is $^1S$ only possible for $n=n'$ and $^3S$ only for $n\neq n'$? There is, unfortunately, no information given about $n$ and $n'$ in the assignment.
And, on a similar note, is the total wave function automatically antisymmetric just because the electrons are in different nuclear shells, leaving the spins free to be orientated in every conceivable way?
 A: 
How can both of these be possible without breaking the Pauli principle?

By adjusting the spatial states occupied by the two electrons, i.e. by changing from $n=n'$ to $n\neq n'$.

My understanding is that the total wave function (consisting of both spin and position-space wave function) has to be antisymmetric.
For $^1S$, the spin wave function is antisymmetric, meaning the position-space wave function has to be symmetrical, while for $^3S$ the opposite would have to be true.

Yes, this is all correct.

Is $^1S$ only possible for $n=n'$ and $^3S$ only for $n\neq n'$?

The second part is correct (a $^3S$ term is only possible if $n\neq n'$) but the first one is only partially so. A $^1S$ term is possible for $n=n'$, but it's also compatible with $n\neq n'$ if the spatial wavefunction is symmetric.

And, on a similar note, is the total wave function automatically antisymmetric just because the electrons are in different nuclear shells, leaving the spins free to be orientated in every conceivable way?

This makes very little sense ─ I can't make heads or tails out of that statement.

And while we're here, a word of warning: combinations with $n\neq n'$ are certainly possible, starting with a $1s\,2s$ state, but you need to be rather careful with them when both electrons are in  $n>1$ shells. To see why, try to estimate (using a hydrogenic independent-electron estimate ─ it isn't great, but it gets the point across) the energy of a $2s^2$ or a $2s^13s^1$ state, say, and compare it to the ionization threshold (i.e. the energy difference from $1s^2$ to $\rm He^+$ in the $1s^1$ state). What does that tell you about that state?
Spoilers:

 Hint, hint, hint. Those links are important reading but it's way better if you work it out yourself first.

