Math Challenge: How many Build combinations in a Guild Wars 8 man team?

Simple question right? How many Build combinations in a Guild Wars 8 man team?
My buddies and I have been running this around for a while now and have come up with some fun numbers. And I'm curious to see how other people would solve for this.

Here's a useful page. http://www.wiki.guildwars.com/wiki/Skill#Skill_quantity
- Don't forget class restriction and Elite restrictions on the skill bar.
- For fun lets assume it's PVE so we have the PvE skills too.
- Any other questions... pick a direction and note the assumption you made.

I have a better one;

How many good build combinations in a party of 8?

Only one answer..

Metaway

I don't know. It's too late at night to do math right now. I'm going to bed.

I do know there are 90^8 different profession combos. 10 Professions combined with 9 secondaries times 8 players. That's 4,304,672,100,000,000 profession combos. Now, are you sure you really want to calculate out the skill possibilities????

Psst, Lacasner. I LOVE the avatar!!!

When you say "builds" does that include attribute points? runes? armor? weapons? mods? levels?

I agree, it's late at night, but I think this is Fuzzy Math. Ten professions and 9 secondaries...but your 8 people can only have the same 10 and 9...so, really only 90 combinations of professions, no matter how many are on the team.

I think you can do this with a pretty simple formula from high school stats:
Since you can't repeat skills (or at least skill names,) it's something like this:

M=max # of skills in game for a character of 2 professions.
E = Elite skills...but team can have only 8 elites...so E=8.

(M) x (M-1) x (M-2) x (M-3) x (M-4) x (M-5) x (M-6) x (M-7) x E x 8.

Everything up to E is the possible combinations for your skill bar. The x8 is your 8-man team.

I think that might end up correct, but maybe I'm forgetting something?

No, they can all be different.

S[ 71+(189/7) + (189/8) x ....] x8
90 times, one for each class
something like that

humm.... so X is....?

Agreed that the original calculations on possible class combinations is incorrect. Remember, the order of the members don't matter. A team with a 6 Mo/Es, 1 E/Me, and 1 W/R is the same in this problem as a team with 1 E/Me, 6 Mo/Es, and 1 W/R in that order.

That makes no difference in this case, as it's a permutation, not a combination. The total amount of class/place combos is still the aforementioned 90^8, since there are no restrictions.

If it were a combination, and order of classes did matter, then it would be 90^8 to the power of 8 factorial (i.e. 40320). Yeah, that's a big number.

If there were restrictions, it would be slightly, slightly more complicated (for example, if you could only have 1 of each primary, the equation would be 90 x 81 x 72 x 63 x 54 x 45 x 36 x 27, for a total of 78,103,970,582,400 combinations)

But factoring the amount of combinations for individual builds would need a super-computer.

Lets see... number of possible skill bars. I'm going to ignore PvE only skills since I don't believe in them.

Now, you could get a most basic idea of how many combos there are by just taking the number of possible skills (1135) and raising it to the 64th power, ignoring professions, elites, skill repeats, and skill orders. That gives you...

3.30929237 × 10^195.

Start with the number of profession combos... 10 * 9 (we'll assume null secondary doesn't matter since it is no different then having a secondary and not using any skills from it).

Now, how many skill bars per profession combo...

First, how many skills are there... 130 skills including 36 elites for the 6 primary classes (close enough! I'm lazy!). There are 100 skills including 25 elites for sin/rit, and 75 skills including 15 elites for P/D. But I'm really lazy, so I'll just take the average... 114 skills including 29 elites. Average, per class. And res sig. This leaves each character with an average of 229 skills (58 elite, 171 non-elite).

So you could make another estimate on skill bars, this time taking into account that each position on the bar really only has 229 options, instead of the full 1135....

229^64 = 1.0702146 × 10^151

That estimate acknowledges that each character has at most 2 classes, but doesn't take into account the 90 different class combos. It also still ignores elite skills and repeated skills.

Now, there are two options that matter for your skill bar... either you use an elite or you don't... these options are additive. So how many skill possibilities per character...

58 elites per two classes and 171 non-elite skills
58*171*170*169*168*167*166*165 (using an elite)
+171*170*169*168*167*166*165*164 (no elite)

Now, take this times 90 profession choices... except we have a problem - a W/R with all warrior skills, and a W/A with all warrior skills are really the same, but counted twice. So you have to remove all the extra copies... for each primary profession 8 of the secondary professions include an extra copy of the 9th secondary profession.

-80 (extra copies) *29*86*85*84*83*82*81*80
-80 (extra copies) *86*85*84*83*82*81*80*79

Now, since in a team of 8 you can have duplicate builds you just have to raise the whole thing to a power of 8 to get the total number of possible teams. This factors in only skills and professions (Mo/E and E/Mo are counted, even with the same build) since they could actually be different, but doesn't double count builds that are actually the same (Mo/W with only monk skills, and Mo/E with only monk skills). It also includes repeats of the same builds but in different orders, but I'm pretty sure that is an insignificant portion of the total number. Individual character builds also have this same issue. The above poster seems to think otherwise... but if you reduce the problem to only 3 skills in 3 skill slots you can see he is wrong - you get 3*2*1=6 combos, but each of them includes the same 3 skills. Anyway, the repeats are more significant in individual builds, but still should be insignificant, probably entirely contained within the rounding (I only gave 8 of 19 digits).

(90*(58*171*170*169*168*167*166*165+171*170*169*16 8*167*166*165*164)-80*(29*86*85*84*83*82*81*80+86*85*84*83*82*81*80*7 9))^8

That gives me a total of the following number of theoretical character builds:

7.51961767 * 10^19

And the following number of possible team builds:

1.02227108 * 10^159 (that's a 1 followed by 159 zeroes, give or take 3 percent)

It would take a crazy amount of math, I think, to eliminate the duplicate builds and teams that only vary order and I really don't think it would have any significant effect.

Actually it's quite complicated.


Look at alone the combinations for 1 single character (example Warrior/Ranger):

- A W/R has a total pool of 257 individual skills (128 W, 128 R + Rezsig).
- You pick 8 Skills out of this pool, that's a combination without repetition and it would be calculated like 257!/(8!-[257-8]!)
- However, that would not be correct, because you can't take more than 1 Elite on a bar, so you have to extract this
- You put an Elite in your bar and calculate the combinations for the other 7 slots:
- One case without all W-Elites and no R-Elites (257-36+35)!/(7!-[257-36+35-7]! and multiply it with 36, to put back the W-Elites again
- The other case would be without the R-Elites and no W-Elites (257-35+35)!/(7!-[257-35+36-7]! ...and mulitply with 35
- Add both cases and you have all combinations for a Warrior/Ranger


Since the classes have all different amounts of Skills and Elites, you have to calculate this for every single of the 90 overall combinations of Primary- and Secondaryclasses and add them. Take care you would have to eliminate the combinations of 1 class only (i.e: W/R and a W/D have all the bars with only Wa-Skills overlapping).
Save some works do it just for the right 45 and muliply them by 2 (because a W/R is the same as a R/W). It's still alot of work, but Excel should help.


After that you would have the number for 1 character, but you can't muliply this with 8, it's as a "Combination with Repetition". Well, there can be muliple W/R on 1 team and all the W/R can have different Skills:


n would be the number you got above, r would be the amount of team-members, in this case 8.

In PvP:

Warrior has 36 Elites, 94 non elite, 1 Rez Sig.
Ranger has 35 Elites, 95 non elite, 1 Rez Sig.
Monk has 35 Elites, 96 non elites, 1 Rez Sig.
Necro has 35 Elites, 97 non elite, 1 Rez Sig.
Mesmer has 34 Elites, 93 non elite, 1 Rez Sig.
Ele has 35 Elites, 99 non elite, 1 Rez Sig.
Assassin has 25 Elites, 75 non elite, 1 Rez Sig.
Ritualist has 25 Elites, 75 non elite, 1 Rez Sig.
Dervish has 15 Elites, 60 non elite, 1 Rez Sig.
Paragon has 15 Elites, 60 non elite, 1 Rez Sig.

Warrior: PvP Skills, ill also add in the chance that they leave a spot blank.

W = 36*96*95*94*93*92*91*90= 2162618283571200 = 2.16 x 10 ^15
W/R = 71*191*190*189*188*187*186*185 = 589103020527159600 = 5.89 x10 ^17
W/Mo = 71*192*191*190*189*188*187*186 = 611393405087646720 = 6.11 x 10 ^17
W/N = 71 * 193*192*191*190*189*188*187 = 634402834311375360 = 6.34 x 10 ^17
W/Me = 70* 189*188*187*186*185*184*183 = 538905834262857600 = 5.38 x 10 ^17
W/E = 71*195*194*193*192*191*190*189 = 682656138980524800 = 6.82 x 10 ^17
W/A = 61*171*170*169*168*167*166*165 = 230292267804799200 = 2.30 x 10 ^17
W/Rt = 61*171*170*169*168*167*166*165 = 230292267804799200 = 2.30 x 10 ^17
W/D = 51*156*155*154*153*152*151*150 = 100034641154448000 = 1.00 x 10 ^17
W/P = 51*156*155*154*153*152*151*150 = 100034641154448000 = 1.00 x 10 ^17
Total = (1+1+2.3+2.3+6.82+5.38+6.34+6.11+5.89+2.16) x 10 ^ ((17*9)+15) = 39.30 x 10 ^168

So if you always have warrior as either primary or secondary you have (39.30 x 10^168 + 37.14 x 10^153)

76.44 x10^321 total combos of Primary/secondary and skills, that covers 19 of the 100 different combos (10 primary * (9secondary + 1 no secondary)

I dont know about the rest of you, but I can't wait to try them all.

I'd say alot.

The answer is 13,142,568

I second that. And now my head hurts.

* looks at the numbers *
* has heart atack and dies *
Nerf the math, plz, it makes me dizzy

It would prolly be easier to calculate the number of french fries eaten every day.

You'll have to use the profession with the higest possible number of skills to get the maximum.

The total number of combinations, counting skills etc, is mind boggling. For those Monty Python fans ..... what is your favorite color?

Blue! No, yel--
*Aaaaargh!*

the answer is 42 :P

actually it's 47 (Trekkie and Alias fan)

For Warrior/Ranger where
- total skills = 189
- total elites = 71

i bet that all possible combinations are something like this:


Which roughly equals to: 1,488 x10^14 possible combinations

I'm not doing it for all other professions :O

Ok, I've had a good night's sleep and 2 cups of coffee. Let's take another look at this.

Ok, how about answering a simpler question first?

How many different PvP skillbars are possible?

Let's start with the lowest numbers first. A dervish or Paragon has 69 regular skills and 15 Elites available to them. I'm deriving that from 60 profession specific skills and 9 skills that any profession can use. (Yes, 9!! Rez Signet, Empty slot #1, Empty slot #2, Empty slot #3........)

For no Elites, that's 69!/(8! x 61!) = 8,361,453,672
With 1 Elite it's (69!/(7! x 62!)) x 15 = 16,183,458,720
For a total of 24,544,912,392 possible skill bars for Dervishes and Paragon using skill from their primary profession only.

For Assassins and Ritualists, it's 84 regular skills and 25 elites. Plugging those numbers into the same formula yields 128,520,753,894 combos for those guys using only skills from their primary.

That's a grand total of 306,131,332,572 skill bars so far and I've only counted 4 professions with no secondary skills. The Core professions have Wayyyy more skills to choose from.

I need more coffee!

well you can't bring empty8 if you don't have empty7, then it would just be empty7

what about all the attribute combinations?
and you could bring different level characters too!
or you could bring less than 8 people

ze possibilities, zey are endless!

Also a simple way of getting elite skills in the equation, and empty skills
slot1*slot2*slot3 etc M being number of non elite skills possible(empty slot not included), E being number of elites

((M)*(M-1)*(M-2)*(M-3)*(M-4)*(M-5)*(M-6)*(M-7) + (M+E)*(M)*(M-1)*(M-2)*(M-3)*(M-4)*(M-5)*(M-6))
+
((M)*(M-1)*(M-2)*(M-3)*(M-4)*(M-5)*(M-6) + (M+E)*(M)*(M-1)*(M-2)*(M-3)*(M-4)*(M-5))
+
((M)*(M-1)*(M-2)*(M-3)*(M-4)*(M-5) + (M+E)*(M)*(M-1)*(M-2)*(M-3)*(M-4))
+
((M)*(M-1)*(M-2)*(M-3)*(M-4) + (M+E)*(M)*(M-1)*(M-2)*(M-3))

you get the idea
i haven't put too much thought into that so it might work. but I think it should
if you look away from the levels and attributes you can put it all into 1 (huge) formula
basically repeat the above for every class combo with different M and E values

I don't know about your formula, haven't looked at it too closely, but you're correct about the empty slots. I realized after the 3rd cup of coffee that I made a mistake with those. The correct formula is bit more complicated.

Dervish and Paragon 61/15 skills (keep the rez sig). Thats (61!/8! (61-8)!) + 61!/7! (61-7!) + 61!/6! (61-6!) + ........ Then repeat for adding back the Elite skills. Should come out slightly less than my original numbers, but still a huge amount of combos.

Ok 4 cups of coffee, 1 giant caffeine buzz and a whole lot of Geek determination and here's the results.



This is only counting builds that use skills from the primary profession only (plus a rez sig). Trying to add Primary/Secondary mixed builds caused the numbers to go up higher than my spreadsheet supports. Feel free to double check the math.

BTW, 10 of these builds are identical (8 empty slots) so you can subtract 9 from that final number.

Now aren't you glad we discarded PvE only skills and Cap Sigs!

Now factor in Attributes and weapons and all that fun stuff.

there is no end to the numbers. when you take into consideration that a "build" consists of insignias, weapons, modifiers,runes,attribute distribution, and skills, the possiblities are impossible to figure out. also you can have the same exact everything that one build has but leave out 1 or 2 things (be it attribute points,or a rune or whatever) and technically that makes it a differant build.i dont think it is possible to figure out.

thetechx, it's perfectly possible to figure it out.

Im only looking at skill bars,
I assume whatever the skill bar is,... the attributes, runes and equipment will support it. Though there will certainly be multipurpose bars, i figure that arguably balances out against the useless bars

For ballpark permutations on 1 character I figure it something down these lines. Please point out any critically flawed logic here.

slot 1 = 1160 possible choices. - picking your first skill you now have a primary class, and if you're going to use an elite, lets assume you've picked it.
slot 2 = 869 possible choices - Except for elites, which would conflict with slot 1, you can still pick any class skill here. you're allowed 2 classes..so if you're going to use a secondary class skill we assume you pick at least one of those skills now.

Ok, now I remove the remaining 8 classes as possible skill choices since we've locked in to our primary and secondary. To simplify I'm going to use an average of 86 non-elite skills per class. 172 choices -1 the non-elite we just picked.

slot 3 = 171 remaining possible choices
slot 4 = 170 possible , slot 5 = 169 possible .... slot 8 = 166 possible.

Which gives me a total of (2.31 *10^19) possible character skill bars.
Multiply 8 to make a full team. (1.85 *10^20) possible team skill bars.
Or, 185 Quintillion. (1 Quintillion = 1 Billion Billions)

??..ballpark since i'm using averages for the smaller (lol) numbers. More accuracy is certainly possible.

You still have to multiply it with 90, because u did only 1 out of 90 possible class combinations

You actually have to make 2.31E19 at the eight power instead of simply multiplying by eight. My first estimation (taking means, without taking into account one-elite only, allowing repetition of skills and such) was about 10^150 posible builds. A second try allowing only one elite and no repeated skills gave 2.7E138. And even so, only three or four are played in PvP.

O________________o




I don't think there's enough "wtf"s in this world to express my feelings.

I thought about trying to figure this out, but as I began to think about it, the shear complexity of the problem made my face turn into one of these.

You guys forget 1 thing, 1 character can't use skills of different attributes,

i.e. Sword warrior can't use Axe and Hammer skills, Pyro ele is unlikely to use spells of 3 other magic, etc etc.

If you factor that in, the equation will get really messy.
Since a War can put 1point in sword, 1 point in axe, 1 point in hammer, etc with weapon swap.

Thus in theory, a sword warrior can use axe/hammer skills, the same can be said by the other classes.

I did it! (Poured the coffee straight into the spreadsheet software this time )

The total number of PvP skill bars possible in the game is 3,682,658,694,659,070. That's 3 Quadrillion, folks! That's a lot of skillbars to test.

Attached is a spreadsheet with all the numbers. In order to attach it here, I converted it to .txt format. The columns are Tab delimited. In order to view it properly, upload it into a spreadsheet program. It's difficult to read in Notepad. Unfortunately, this destroyed the formulas that were in each spreadsheet cell and left only the resulting values. If you really want to see the formulas, contact me and I'll e-mail the spreadsheet to you. (Sorry, I couldn't attached an .xls file here. Security precaution since they can contain macro viruses)

skillcombos.txt

So, is there a prize for this? Other than the Guru Geek of the Year award that is.

I started by counting up the skills available for each profession in PvP. If my counts are wrong, please reply. +1 has been added to all professions for the Rez Sig.
Warrior 95/36
Ranger 96/35
Elementalist 100/35
Mesmer 94/34
Necromancer 98/35
Monk 97/35
Assassin 76/25
Ritualist 76/25
Dervish 61/15
Paragon 61/15


What's included and what's not:
- No PvE only skills were added. By doing this I was able to drop the Cap Sig which would have complicated matters even further since you can have more than 1 on the bar.

- No Eye of the North skills

- This is evaluating unique skillbars only. Runes, weapons, armor, attribute distribution is not considered. (No!! I'm not calculating that. Don't even ask!!!!!!! )

- There are 55 different profession combos in the game. For these purposes, a Warrior/Ele is treated the same as an Ele/Warrior. Both can run the same skillbars. A Warrior with no secondary though is a unique combo.

- Order of skills doesn't matter. A bar with Spiteful Spirit, Arcane Echo and 6 empty slots is the same as a bar with Arcane Echo, 2 empty slots, Spiteful Spirit and 4 empty slots and is only counted once.

- Duplicate bars across professions was eliminated in the final total.
For Example:
A Dervish has 10,917,708,417 different skillbars available using only Dervish skills (and a Rez Sig).
A Paragon also has 10,917,708,417 skillbars.
A D/P has 2,982,677,338,196 skillbars, but 21,835,416,834 of those are duplicates of the Dervish only and Paragon only bars. That 21 billion was subtracted from the D/P total for the final numbers.
Another Example: A warrior has 131 skillbars that use only 1 slot. A Ranger also has 131 bars. A Monk has 132. For a total of 394. Wrong. 3 of those bars are identical. (Rez Sig + 7 empty slots) For this reason 54 was subtracted from the total.

- ALL PvP bars are counted. The good bars, mediocre bars, bad bars and the truly horrible you-should-uninstall-the-game bars. (Flare, Mending, Riposte, Cyclone Axe, Hammer Bash, Deep Freeze, 2 empty slots) I don't judge 'em. I just count them.

- Oops, I lied. Not every PvP skills was counted. Dragon Arena skills, Snowball skills, Beetles and any other speciality arena skill wasn't counted.

Whew, I'm outta coffee. I'm going home.

Edit: Oops, almost forgot. Multiple that number (3,682,658,694,659,070) by 8 to get the answer to your 8-man team question.

Lock this before mah brain explodes...