Customization (Numbers)

(Sorry for the TL;DR post. I put the important numbers in bold, so many of you may just skip to that, and find the numbers interesting. I start out with the parameters and assumptions I'm making. I then give the relevant numbers. Afterwards, I explain my calculations for those who think I'm blowing smoke, or are just generally interested [I hope that you read them even if you believe me] But at least you can look at the bold numbers.)

I don't know if it's been done, but I thought it would be fun to mention the actual numbers for how many possible specializations you can give a character, excluding gear; only accounting for skills, and traits activated, and runes. Since there's never a good reason to have a skill without a rune (there will always be at least one which just flat out benefits it with no compromise), I'm excluding the possibility of having skills without runes. I also won't account for the fact that different tiered runes can be inserted, as we'll always mechanically aim for the highest tier rune.

I'm assuming a class has 24 skills and 20 traits. 6 skills and 3 traits may be activated, and each skill can be socketed with one of five types of rune. With that, there are 2.4 trillion specializations (2.397 x 10^12).

If you exclude passive traits, that gives you 2.1 billion (2.103 x 10^9).

Let's look at a slightly more realistic model. I remember on the old bnet forums, that a mod mentioned that on average, each class should have something like 12 skill types of a particular nature, and 2 of each type. With that, it wouldn't be ideal for a character to have two of the same type. (For example, both Slow Time and Wave of Force act as a crowd control mechanism for a wizard, and with only 6 active skills, you probably won't find a wizard with both). With this model as an approximation, the new numbers are:

1.1 trillion (1.053 x 10^12) including traits, and nine hundred million (9.24 x 10^8) excluding traits.

Interesting, right? For those of you who don't know how to get those numbers, hopefully you were thinking "where the hell did he get these", rather than "oh okay, cool thanks". If you know how to get them, you'll probably be very bored by the following, so you can just skip to the last paragraph, or stop here.

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If you have N things to choose from, and you can select R of them, and you don't care about the order in which you pick them (we don't care about the order in which we pick skills), then the number of ways you can make this choice is

N Choose R = N! / [R! (N-R)!]

Where an exclamation point is a "factorial"; N! = Nx(N-1)x(N-2)x.......x2x1. (2! = 2, 3! = 6, 4! = 24, etc...)

Here's why. In your first choice, you have N things to choose from. In your second choice, you have N-1 things. Then N-2. On your R'th choice, you have N-R+1 things to choose from. So the number of possible ways to choose them is

Nx(N-1)x(N-2)x.....x(N-R+2)x(N-R+1) = N!/(N-R)!.

However, in this calculation, we cared about order. That is, counting skill A and then skill B was a different outcome from skill B and skill A. With R skills, there are R! possible ways you can order them, so with N!/(N-R)!, we counted each set of skills R! times (each with a different order), so we divide by R!. Thus, there are N! / [R! (N-R)!] possible ways of picking R objects out of N, where the order doesn't matter.

So, with 6 out of 24 skills, and 3 out of 20 traits, we already have a customization of (24 Choose 6)x(20 Choose 3). That is excluding runes.

Now if you have R objects, and each object has S manifestations (eg. you have 6 skills, each has 5 manifestations from a different rune), then you have S^R (S to the power of R) possible ways of manifesting those R objects. Here is a proof by induction.

1) If you have one object, and there are S manifestations of each object, then there are only S ways of manifesting that one object.

2) Say that with N-1 objects, there are k ways of manifesting those objects. Well, if you add one more object, with S ways of manifesting that one object, then there will be k manifestations of the whole set for each manifestation of that one object. Therefore, with k ways of manifesting N-1 objects, there are kxS (k times S) ways of manifesting N objects.

Apply statement 2) to statement 1), where N = 1. That says that one object has S manifestations so two objects must have S� manifestations. Apply statement 2) to that, where N = 2, and you get that 3 objects has S^3 manifestation. Apply statement 2) N times, and you get that there are S^N ways of manifesting N objects. Proof by induction.

In our context, with 6 skills, and 5 ways of manifesting each skill, there are 5^6 = 15625 possible ways to rune a particular set of 6 skills (remember, excluding skills that have no rune).

So objectively, there are (24 Choose 6)x(5^6)x(20 Choose 3) ~= 2.4 trillion possible ways to select 6 skills, put runes in those skills, and select traits.

Remember my approximation to eliminate builds that aren't viable. If each character has 12 types of skill, with 2 of each type, and you would never pick two of the same type, then the calculation becomes (12 Choose 6)x(10^6)x(20 Choose 3) ~= 1.1 trillion builds.

That is, you picked 6 out of 12 types of builds, so 12 Choose 6; then each of those 6 types had two skills to choose from, so 2^6, and each skill had 5 possible runes, so 5^6 (note 2^6 x 5^6 = 10^6; another way of thinking about it is that instead of two skills each with 5 types, you really just had 10 different skills to choose from for each type), and finally 20 choose 3 for traits.

Discussion! For example, maybe my estimation of 12 types of skills with 2 of each type was a bad estimation (although I can swear I saw something like that in a blue post once; I may try to find it later). Or maybe the number of skills and traits that I was assuming (24 and 20) is not the same, or inconsistent with character class (that's what the Wizard is given). In that case, though, it's easy to just redo the calculation with a given number of skills and traits. Maybe there will be lots of runed skill combos that simply aren't viable in practice, even if the skills are of different type.

EDIT: Okay, I just found the blue post to which I was referring.

http://forums.battle.net/thread.html...=2&sid=3000#27

Just read the paragraph right after that whole dice list.