Classes with spells or maneuvers have thousands, if not million ways of combining powers. But what about the classes that only have talents? How many different (theoretic) builds are there?
It turns out that just by Core Book / SRD, the Paladin is the class with the fewest avaiable talents (9), followed by Barbarian (10) and Ranger (12). The Druid has even less, just 6.
However, to figure out how many different build combinations there are just by talents, we need a bit more math. The key element here is from combinatorics, called (X choose Y), usually written in brackets, with X on top and Y below. The idea is that you have a pot with X numbers, and you pick Y numbers from it. (X choose Y) gives you the possible unique sets that you can draw, when the order in which you draw them does not matter. A quick way to calculate this is to enter something like "10 choose 3" into Wolfram Alpha. This gives you the number of possible builds for a class that chooses 3 talents at level 1 from a list of 10.
To keep the maths easy, I will look only at talent choices. Of course, race, feats, icon relations, backgrounds, and choices within a talent (such as which spell for Ranger Ex Cathedral, or which bonus for Warrior Druid) create enough variety that two PCs with the same talent set can be very different. And that's before we have even started with the One Unique Thing.
The Barbarian has his 10 talents split into 6 Adventurer / 2 Champion / 2 Epic.
At first level, that's (6 choose 3) = 20
At level 5, you can take any of the three remaining adventurer talents or one of 2 champion talents, giving you 5 choices.
At level 8, you can take any of the 10 talents, but you already have 4 on your character sheet, for 6 remaining options.
(6 over 3) x 5 x 6 = 20 x 30 = 600
600 Core Book Barbarian builds.
Leaving out Way of Evil Bastards for now, we have 8 Paladin talents.
You choose 5 talents over your career. For a Paladin starting at level 8, the level at which he takes each talent does not matter. In that case, there are (8 choose 5) = 56 builds.
Let's put Way of Evil Bastards back in.
Of those 56 options, if we require that you choose Path of Universal Righteous Endeavor first, the there are (7 choose 4) = 35 remaining build options. For these 35 builds, we can switch PURE for WEB. So we're looking at 56 + 35 = 91 possible level 8 Paladin builds.
However, you choose 3 talents at first level and then one at 5th and one at 8th level. If we want all builds, accounting for that difference, we need to look at how many ways we can order 5 talents that way. Out of your 5 talents you choose for one build, you pick one of 5 for champion, and then one of 4 remaining for epic - the remaining three are adventurer so no choice left - for 20 options.
91 sets with 20 versions of each, that's...
91 x 20 = 1820
Leaving Animal companion aside, that's 11 talents. If you start at level 8, that's (11 choose 5) = 462 builds.
As above, there are 20 ways to sort each set, for a total of (11 choose 5) x 20 = 462 x 20 = 9240 builds.
For the animal companion, you would normally take it at 1 or leave it. If you take it, you then take 1 from the remaining 11 talents at levels 1, 5, 8.
11 x 10 x 9 = 990. In total, 10230 build options.
Now you know...
The Druid has only 6 talents, the least of any class, but all of them allow to "double up". So he can either choose 3 Initiate talents or one Initiate and one Adept.
3 Initiate: (6 choose 3) = 20
Adept / Initiate: 6 x 5 = 30
20 + 30 = 50 Druid builds.
This class is a spellcaster, but doesn't choose any spells (the class spells are from a fixed list and the "stolen" spells are random). So it's a straightforward case of choosing 3 from a pool of 8.
(8 choose 3) = 56
Just looking at talents doesn't do the Bard any justice. Not only does he get the widest choice of powers (6 battle cries and 7 spells / songs), he has multiple talents that grant extra powers, including from other classes. But this is out of scope here.
There are 8 bard talents, but both Battle Skald / Spellsinger and Loremaster / Mythkenner are exclusive. (8 choose 3) = 56.
There are 6 builds for Skald + Spellsinger + any that we need to remove, and 6 for Loremaster + Mythkenner + any. So that's 56 - 12 = 44 bard builds.
But what about battlecries and spells?
At level 1, you start with (6 choose 2) = 15 choices for spells / songs and (4 choose 2) choices for battlecries.
At level 10, this massively increases to (18 choose 7) = 31,824 combinations of songs and (11 choose 6) = 462 combinations of battlecries.
Talents are straightforward. (10 choose 3) = 120 combinations of domains.
For spells, at level 1 you get (7 choose 4) = 35 combinations at level 1 and at level 10 (20 choose 9) = 167,960 combinations for spells (ignoring that you further need to pick one you'll only know at level 7.
8 talents in total, of which you choose 3 at level 1 and 1 at level 6. So that's (8 choose 4) = 70 builds, or 70 x 4 if you make a difference in which talent you pick at level 6.
At level 1, you have (10 choose 3) = 120 variations in what maneuvers you can pick, which goes up to (23 choose 8) = 490,314 at level 10.
Only a small selection of these combinations will be a viable build though, as you need to make sure you cover all possible natural attack rolls (odd/even, hit/miss)
Straightforward (8 choose 3) = 56 possible builds with talents.
Attack powers give you (6 choose 4) = 15 options at level 1, which open up to (17 choose 9) = 23,410 at level 10.
(9 choose 3) = 84 combinations of heritages and other talents.
(6 choose 4) = 15 spell combinations at level 1. (20 choose 9) = 167,960 spell combinations at level 10.
Talents: (7 choose 3) = 35
For spells, I'll assume the smart choice of reserving one slot for the general utility spell. While you can cast any spell from the utility list that way, you don't need to choose it at character creation. So we'll only choose spells for the remaining slots. We'll further assume that these will be used for combat spells, instead of more utility spells.
Spells at level 1: (8 choose 4) = 70
Spells at level 10: (29 choose 11) = 34,597,290
You gain 4 talents in total, the 4th at level 5. That's (12 choose 4) = 495. If you care which talent you choose at level 5, that's 495 x 4 = 1980 build combinations. Note that this includes builds with both Strategist and Tactician. You can take both talents at the same time, but it's not particularly useful.
Powers at level 1: (9 choose 4) = 126; Powers at level 10: (30 choose 10) = 30,045,015. Note that it is suggested to pick a mix of commands and tactics with this class, which only makes a smaller part of those 30M possible builds useful.
Other classes: Monk, Necromancer, Occultist.