I am considering the possible Wick contractions for the following expression: \begin{align*} \left< x ( t' )^5 x ( t'' )^5 \right> = \left< x( t' ) x( t' ) x( t' ) x( t' ) x( t' ) x( t'') x( t'') x( t'') x( t'') x( t'') \right> \end{align*} I believe there are a possible of $(10-11)!! = 9!! = 945$ possible contractions to be consider. I think there are three possible classes of diagrams to consider (see the attached figure):
(a) Connect each $x(t')$ to each $x(t'')$ which gives a total of $5!=120$ contractions
(b) Connect two $x ( t' ) $ to each other which implies the remaining three $ x ( t') $ are connected to three $ x ( t'' ) $ and the remaining two $ x( t'' ) $ are connected to each other, with multiplicity $\!^5C_2 \times 3 ! \times \!^5C_2 = 10 \times 6 \times 10 = 600$
(c) Connect four $x ( t' ) $ to each other which implies the remaining $ x ( t') $ connects to one $ x ( t'' ) $ and the remaining four $ x( t'' )$ are connected to each other, with multplicity $\!^5C_4 \times 1 ! \times \!^5C_4 = 5 \times 1 \times 5 = 25$.
However, this can't be correct since this gives a total of $120 + 600 + 25 = 745$ contractions which is not equal to the expected number of $945$ contractions. Any suggestion where I went wrong?
