Maths problem, combinations

I’m struggling with one of the follow up questions from the senior maths challenge and was wondering if the answer was 192 or 384 or something else?

Okay I did it again to show you my working out.

I can’t really explain it. It’s a slightly exhaustive method. I chose examples of numbers for a given colour arrangement and looked at the amount of choices of Fnargs I had for each number I chose in the next step (this is what my numbers CX means, the x is the amount of number of Fnargs I could chose).
I had to split off a couple times when some of the different choices of heads limited the next choice, so I worked out the choices for each individual choice following on.

There is probably a better way to do it, and im pretty sure you won’t be able to follow my method :P

One proof is similar to the 2015 Q16.
Just draw a 2*4 grid for colour/heads and multi;ly the number of possibilities at each stage.
You should be able to work out which one of you is right fairly easily.
If not, post your working.

I’m confused what you mean by this? This is kind of what I attempted, but when I was doing it some possibilities limited the next options so I had to split it into multiple ‘branches’ almost.

I didn't really come across any significant branching. The first two were always 8 and 3 and there were three other 2s. When you get down to the last three choices, they're unique (1s).