Assume that everything that matters to us is what you get directly from being a fighter. The fighter takes four categories of decision; Let's cover them one by one.
As you say, Fighting Style is chosen in level 1, which gives 5 options
On the first level, wrestlers choose two skills from a list of eight, which gives 28 Total options
First there is the Champion. At level 10, the champions get a second fighting style; it seems that there should be four options, but since we do not care in what order we get our Styles, there are only ten possible pairs, so the multiplier for this is two.
The next one is the Battlemaster. They choose five maneuvers per level 12, from a list of 16; that's 4368 possible options They also get competition in one of 17 Craft tool sets, for a total of 74256 Battle Masters.
Finally, the Eldritch knight. By level 12 they know 8 spells; they can be any combination of magician spells of 1st and 2nd level, and 6 of them must be Abjuration or Evocation. There are 15 spells that satisfy all these conditions and 49 satisfy all, except the requirement of the School. Then they can have (15C6 * 49C2) + (15C7 * 49) + (15C8) = 5005 * 1176 + 6435 * 49 + 6435 = 6207630 spell choices Added to that 3 of 16 cantrips, a total of 560 Cantrip options, and you get 3476272800 possible gentlemen of Eldritch.
Add the classes and you get 3476347058.
How is it
A level 12 fighter has 4 ASIs; if all are spent as ASI, then the player has 8 points to distribute more than 6 skills, which is 13C5 = 1287. If 1, 2 or 3 are spent as ASI, then the total number of options is 21, 126 or 462. Cool.
The player's manual offers 42 feats; They are also incredibly complicated. Maybe later I'll edit in an analysis of what individual feats do to mathematics (spoiler: it's very complicated), but for now I'm just going to pretend that each feat can be taken once and does not require decisions. So for 1, 2, 3 or 4 feats, you have 42, 861, 11480 or 111930 elections Cross-multiply these numbers, and you come across 2350530 + 1446480 + 397782 + 54054 = 4248846 options here
Put these options together and you will get: 5 * 28 * 3476347058 * 4248846 = 2.0678649e + 18. That's just a hair in two quintillions of options.
You state in the comments that "This would be a useful point for subsequent discussions of "everything is the same" and "feat / skill trap""This is wrong, I would not do it, two quintillion is a nonsense number, list the following useful character options:
- A fighter with 14 Charisma and 15 Wisdom, and the same character with 15 Charisma and 14 Wisdom.
- A wrestler who took the Endurance (Strength) feat for +1 Strength and no other effect, a Hunter who took the Master of Arms feat for +1 Strength and no other effect, and a fighter who took the feat Very Armed for +1 Force and no other effect
- 1364 Battle Masters who only use Precision Attack and ignore their other four maneuvers.
- An Eldritch knight who can throw Light, and one that can launch Dance lights
- A champion who put the 8 points of ASI in Intelligence.
And each of these characters overlap and repeat themselves millions of times, along with many others. I have not done anything useful here; I've made a fun math problem, and this answer does not deserve a place in any half-hearted conversation.