I recently learned about the Enigma Machine in my cryptography class, but I am a bit confused as to the number of permutations of the wheel settings. According to every article I've read on the matter, the number of different ways the wheels could be set up equals the number of permutations that the wheels can produce. I understand that there are 26*26*26 possible starting settings (assuming three wheels), but what I do not understand is why the settings of each wheel do not come into play. From what I know, each wheel contains a series of jumbled wires that further scramble each incoming letter. Aren't there 26! different ways that these wires can be arranged? For example, when the plaintext is an 'A' and the starting setting of the first wheel is at position n, the first wheel can produce any different letter depending on the wiring inside of the wheel. I just don't get why that is not a factor when determining the number of keys that can be used. Can somebody clear this up please?
EDIT: to clarify, I know that the wiring settings remain constant throughout the encryption procedure, but there are still multiple ways to set it up initially. At this point I'm thinking that each wheel has a specific wiring that is common to all Enigma machines. That would make the most sense to me.