# How to find the number of distinct contraction cases in Wick's Theorem?

Let $\mathcal{G}^8_{un}:=(t_1,t_2,t_1'^3,t_2'^3)=\langle 0 \mid T[Q_{un}(t_1)Q_{un}(t_2)Q(t_1')^3Q(t_2')^3] \mid 0 \rangle_{un}$

We want to use Wicks theorem to write this function as the sum of 2-point functions. Apparently there are eight different types of contraction. Some examples are such as:

(and there are 5 other cases).

Can you determine the number of distinct contractions for a general n point function before you go through Wicks theorem by brute force? This would provide a good of checking that you have covered all possible cases when using Wick's Theorem.

For reference see page 14 of these notes http://www.maths.bris.ac.uk/~dc13950/lecturenotes.pdf