I'm currently studying Wick's theorem for fermions with Peskin's and Schroeder's Introduction to QFT (p.115 & p.116). Here, Wick contractions are defined as
$$
\psi^\bullet (x)\bar{\psi}^\bullet(y) = \begin{cases}\ \ \{\psi^+(x), \bar \psi^-(y)\} & \text{if } x^0 > y^0\ , \\ -\{\bar \psi^+(y), \psi^-(x)\} & \text{if } x^0 < y^0\ \end{cases} \tag{4.105}
$$
and one can show that this coincides with the Feynman propagator $S_F(x-y)$. So far so good.
Then, it goes on to state Wick's theorem for fermions as
$$
T\left[\psi_1\bar{\psi}_2\psi_3 ...\right] = N\left[ \psi_1\bar{\psi}_2\psi_3 ... +\text{ all possible contractions} \right],\tag{4.111}
$$
where $\psi_i \equiv \psi(x_i)$ and spinor indices where neglected.
I'm a bit confused by what is meant with 'all possible contractions' as the contraction was only defined for the ordering $\psi^\bullet (x)\bar{\psi}^\bullet(y)$ and not for $\bar{\psi}^\bullet(x)\psi^\bullet(y)$. How would one define the latter or rather is it necessary to define it at all?
Consider for instance the vacuum expectation value
$$
\langle 0 \vert T \psi_1\bar{\psi}_2\psi_3\bar{\psi}_4 \vert 0 \rangle
$$
such that only fully contracted expressions remain. How would one apply Wick's theorem in this case?
My guess would be to anticommute $\bar{\psi}_2$ and $\psi_3$ and evaluate the resulting expression, i.e.
$$
\langle 0 \vert T \psi_1\bar{\psi}_2\psi_3\bar{\psi}_4 \vert 0 \rangle = - \langle 0 \vert T \psi_1\psi_3\bar{\psi}_2\bar{\psi}_4 \vert 0 \rangle + \delta^3(\mathbf{x}_2 - \mathbf{x}_3)\delta^{ab} \langle 0 \vert T \psi_1\bar{\psi}_4 \vert 0 \rangle,
$$
with $a,b$ being the spinor indices of $\bar{\psi}_2$ and $\psi_3$.
In this case one could simply apply the contraction definition from above, minding that one gets additional factors of (-1) for contractions of not neighbouring operators.
Could someone confirm this approach and/or elaborate more on this?
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