An analysis: Number of builds for each class.

[Drsniper]

I posted this on the official forums. I am posting here as well. Let me know what you guys think.

So I am pretty sure lots of people have asked this question as to how many possible builds are there for each class in D3 (Elective mode enabled). I didn't find the answer anywhere myself, so I decided to create a thread on it. Here's my take on it.

A build consists of any 6 active skills and any 3 passive skills.
For each class
# of active skills = n
# of passive skills = p
Each active skill has 5 runes

No. of ways of picking 6 active skills from a choice on n = nC6. (n "Combination" 6)
No. of ways of picking a rune for each of these active skills already chosen = 5 x 5 x 5 x 5 x 5 x 5 = 5^6
No. of ways of picking a passive skill for a choice of p = pC3. (p "Combination" 3)

Total No. of possible builds for each class, S = nC6 x 5^6 x pC3.

nC6 = n!/(6!(n-6)!) and pC3 = p!/(3!(p-3)!).

Now let's compute the no. of skills for each class individually.

Barbarian

# of active skills = n = 22
# of passive skills = p = 16

Therefore S = 22C6 x 5^6 x 16C3 = 652,863,750,000. (652 billion+)

Demon Hunter

# of active skills = n = 23
# of passive skills = p = 15

Therefore S = 23C6 x 5^6 x 15C3 = 717,670,078,125. (717 billion+)

Monk

# of active skills = n = 21
# of passive skills = p = 14

Therefore S = 21C6 x 5^6 x 14C3 = 308,626,500,000. (308 billion+)

Witch Doctor

# of active skills = n = 22
# of passive skills = p = 15

Therefore S = 22C6 x 5^6 x 15C3 = 530,451,796,875. (530 billion+)

Wizard

# of active skills = n = 25
# of passive skills = p = 15

Therefore S = 25C6 x 5^6 x 15C3 = 1,259,070,312,500. (1259 billion+)

Things to note
Wizard has the most no. of builds and Monk has the least. Wizard has more than 4 times the no. of builds as the Monk.Some interesting comparisons involving the no. of Monk builds since it has the lowest no. of builds.
► If every person in this world (~7 billion) plays Monk in D3, then it is possible to play in such a way such that every single person is playing with a different build.

► So if every single person in this world is playing Monk in D3 with a different build, it will still only cover 2.27% of the all the possible Monk builds.

► So will there be a time when it is certain that 2 people will have the same build?
Yes. Going by the present growth rate of world population (1.1%), the population of Earth will reach 310 billion in the year 2569. If every person in the world is still playing Monk in D3 at that time, then it is guaranteed that there will be atleast 2 people who have the same build

► How much time will it take me to test all the builds for Monk?
If you test a build each second, then it will take 9787 years to check all the 308billion+ builds for the the Monk.

Since checking builds every second isn't possible, so let's say you check a new build every 5 minutes. Then it will take 2.9+ million years to check every build for the Monk.

► And we haven't even touched the other 4 classes which have a lot more builds than the Monk.

http://us.battle.net/d3/en/forum/topic/4768346902

[Hellpyre]

First off: Nice. I enjoyed reading it.

But I think your final bit on the time it would take only works if you disregard redundancy - I.E. I already know how this skill works with this other skill. Needing to tied only to an immediately proceeding and following skill will probably reduce the time needed. (Also, you didn't account for no rune as a rune choice, perhaps for aesthetic reasons on the part of the player)

[galzohar]

If you really want to get meaningful numbers you must cut down at least some of the useless builds.

Builds with no generator/free skills are mostly pointless. So are builds that can't spam any resource spender. These 2 restrictions alone cut down a huge number of builds.

Then you have the fact that it would probably be quite inefficient to have 2 generator/free skills or 2 spam-able spenders, so that cuts down some more builds.

Then again, the above rules aren't set in stone, but I just don't see anyone sensible breaking them except for some extreme special cases for a select few builds. But then again those "select few" builds are actually a lot of builds when you keep in mind that even if you pick a very specific spender and 2 very specific generators, you still have 3 non-spender-non-generator skills to choose from and those add a lot of combinations.

To sum it up, getting total number of viable builds is quite difficult, but if you want the estimation to be more reasonable, you might want to stick with those restrictions and then see what numbers you get.

[FireIceTalon]

And these numbers dont even take gear or your followers (such as the Templar) into account....rofl.

[yovargas]

If you really want to get meaningful numbers you must cut down at least some of the useless builds.

Builds with no generator/free skills are mostly pointless. So are builds that can't spam any resource spender. These 2 restrictions alone cut down a huge number of builds.

Then you have the fact that it would probably be quite inefficient to have 2 generator/free skills or 2 spam-able spenders, so that cuts down some more builds.

Then again, the above rules aren't set in stone, but I just don't see anyone sensible breaking them except for some extreme special cases for a select few builds. But then again those "select few" builds are actually a lot of builds when you keep in mind that even if you pick a very specific spender and 2 very specific generators, you still have 3 non-spender-non-generator skills to choose from and those add a lot of combinations.

To sum it up, getting total number of viable builds is quite difficult, but if you want the estimation to be more reasonable, you might want to stick with those restrictions and then see what numbers you get.

A good way to do this might be to see how many builds are possible without Elective mode since the those builds tend to be very reasonable - 1 generator, 1 spender, 1 defensive and they remove stuff like multiple mantras or picking multiple 2min CD skills. They're not perfect, of course, but I'd be curious see what the non-Elective mode numbers would be.

[galzohar]

That would be pretty easy to calculate. Just replace the nC6 in the formula with multiplying the number of possible skills (3-5) in each of the 6 slots. For example, for the barb that would be 3*4*4*4*4*3.

However, most builds I would consider using (if not all of the ones I came up with so far) require using elective mode.

[londonereight]

For the curious, non-elective mode builds by class (not counting passive skills or runes):
Barb: 3*4*4*4*4*3 = 2304
DH: 4*4*3*4*4*4 = 3072
Monk: 4*3*4*3*3*4 = 1728
WD: 4*4*4*3*4*3 = 2304
Wiz: 4*4*4*5*5*3 = 4800
To add runes, I assume all you would need to do would be to multiply each of those factors by 5 (I'm not counting unruned skills as an option). That gives you:
Barb: 15*20*20*20*20*15 = 36,000,000
DH: 20*20*15*20*20*20 = 48,000,000
Monk: 20*15*20*15*15*20 = 27,000,000
WD: 20*20*20*15*20*15 = 36,000,000
Wiz: 20*20*20*25*25*15 = 75,000,000
Depending on the level of specificity you want, that should give you a rough heuristic for total viable builds, not counting passive skills. I'm not going to do the math on that, partly out of laziness, partly because I'm not sure that every passive skill will necessarily pull its own weight depending on the skills you choose. For instance, the wizard has several passives which are only useful if you deal a certain kind of damage. If the build you rolled didn't have that kind of damage, you would never select that passive.
Right, that's all for tonight, then.

[Karpalo]

Swapping skills between left and right mouse button hardly labels as a new build. In that sense that isn't totally correct.

[Alutemu]

Swapping skills between left and right mouse button hardly labels as a new build. In that sense that isn't totally correct.

In this same vein of thought, these calculations deal with total number of combinations; however, it does not exclude left/right-mouse button swaps nor 1-4 numbered keys swaps.

Let's take the Barb's first 3 skills:
For simplicity's sake, Skill 1 (S1), Skill 2 (S2), Skill 3 (S3).

For these previous calculations, these are all viable builds: S1-S2-S3, S1-S3-S2, S3-S1-S2, S3-S2-S1, S2-S1-S3
I will admit that I'm not a math specialist, but is there a way to cull these down to account for this type of redundancy?

[galzohar]

Yes, there is. There are well-known formulas for selecting N elements out of M elements when order of selection doesn't matter, and another for when the ordering matters.

The correct number of builds is:

(N!/(6!*(N-6)!)) * (M!/(3!*(M-3)!)) * 5^6



First part is choosing 6 skills out of N skills, 2nd part is for choosing 3 out of M passives, 3rd part is choosing 1 out of 5 rune options for each of the 6 skills,

Of course, you still need to cut down stupid builds like builds with no generators or spenders and the likes, at least the ones that have no chance of working. Of course, there are also builds with no defensive abilities or stuff that just flat out suck together, so there's really no end to this.