# Srednicki's QFT: Feynman Rules and Feynman Diagrams

I'm reading Srednicki's Quantum Field Theory. I 'm trying to read Srednicki's presentation of Feynman Diagrams in the chapter Path Integral for the Interacting Field Theory.

The path integral for the $\phi^3$ theory is:

$$\sum_{V = 0}^{\infty} \frac{1}{V!} \Bigg [\frac{i Z_{g} g}{6} \int d^4 x \; \Bigg (\frac{1}{i} \frac{\delta}{\delta J(x)} \Bigg )^3 \Bigg ]^V \times \sum_{P = 0}^{\infty} \frac{1}{P!} \Bigg [\frac{i}{2} \int d^4 y \; d^4 z \; J(y) \triangle(y - z) J(z) \Bigg ]^P \tag{9.11}$$

I get the following:

1. Feynman Diagrams are a away to to organize the terms in the aforementioned mammoth of an expression;
2. I understand the rules. See Srednicki for more details.
3. A diagram may represent a lot of different terms -- that is, those terms would be equivalent. That factor is given by the term: $V!P!(3!)^P(2!)^V$
4. Note that the coefficient from the Taylor Expansion is: $\frac{\displaystyle 1}{\displaystyle V!P!(3!)^P(2!)^V}$. It seesm our counting factor exactly cancels the Taylor Expansion coefficient. Let's say that the numerical factor, after cancellation is, 1. But we may have over counted -- that is, a combination of permutations, described in the text, gives the same diagram. This is called the symmetry factor of the diagram. So, we must divide the numerical factor by the symmetry factor.

My question is as follows:

Given my understanding of the Feynman Rules and Feynman diagrams, I am not sure how to figure out how many diagrams correspond to a fixed values of $V$ and $P$, say $V = V_{0}$ and say $P = P_{0}$. Let's say I have made a diagram, and I have computed its symmetry factor. I'm not sure how to figure how do I know how many different other diagrams are there and when have exhausted all possibilities.

It'd be great if someone could help me on this front.