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?