How can the symmetry factor for a diagram be quickly determined and verified?

[ChrisVer]

Can someone check the symmetry factor I've found in the following diagram and verify it?
Do you know if there is any way to determine the SF of a diagram fast?

So I have $\phi_x$ which can be contracted with 4 $\phi_w$ : 4
Then $\phi_y$ which can be contracted with 3 $\phi_w$ : 3
Then I have 2 $\phi_w$ which can be contracted with 4 $\phi_u$: 2x4
Then I have 3 $\phi_u$ which can be contracted with 4 $\phi_z$: 3x4
The rest $\phi_u$ get contracted together (only 2 left), whereas the $\phi_z$ we get a factor of 2 since there are 2 possibilities to contract 3 fields.
Finally the last contraction gives just a factor 1 (no possible alternative choices).

So the result from contractions is 4x3x2 x4 x3 x2 x 4 = 4! x 4! x 4
The diagram is of order 3 (3 vertices) so there is a factor 1/3! and also the 1/4! from the coupling constant for $\phi^4$ theory.
So is it correct to say that the symmetry factor is afterall SF=16?

 

[nrqed]

Your factor of 2 (in blue) should be 3. From three fields, there are three possible distinct pairs to construct (AB, AC and BC).
But there is also a second point: there should be a factor of 1/4! for each vertex, so you should have 1/(4!)^3 instead of 1/4!
As for a shortcut: usually people drop the denominator and simply calculate the symmetry factor of the graph (the number of automorphisms in mathspeak). I think there is a nice discussion in Peskin and Schroeder if I recall correctly.

 

[ChrisVer]

In Peskin they say that they drop the 1/4! factor and write the vertex factor as $\int d^4 z (-i \lambda)$ because they say that to a generic vertex has four lines coming in from four different places so the various contractions of $\phi \phi \phi \phi$ is 4!...Also the n! from the Taylor expansion will cancel because of interchanging of vertices...
But I don't understand either of these explanations...
So in case I used (1/4!)³ I would have obtained the correct result?

$A= \frac{16}{4! 4!}= \frac{1}{3! 3!} = \frac{1}{36}$
So the symmetry factor is: 36?