In case of superposition of identical particles, we usually just add their amplitudes. For example, if we have several particles having the amplitudes of being in a particular quantum state $\psi_1, \psi_2, \psi_3 ...$, after superposition, we can say, the amplitude of finding one or more particle in that state is $\psi_1 + \psi_2 + \psi_3 ...$, and the probability is $|\psi_1 + \psi_2 + \psi_3 ...|^2$.
Now, my question is, does this superposition work for fermions? I mean, Pauli exclusion principle states that, more than one identical fermions cannot be in the same state at the same time together. Should the amplitude of finding one particle in that state still remain $\psi_1 + \psi_2 + \psi_3 ...$? Or it will be something less than that?
Moreover, since bosons 'like' to stick together, should that change the total amplitude in case of bosonic superposition? Should the total amplitude of finding one or more bosons in a particular state be more than $\psi_1 + \psi_2 + \psi_3 ...$?
Thanks in advance.