Research How many possible teams are there in Pokemon ? A comprehensive and detailed answer.

[Delibird Heart]

Hello everyone, Delibird Heart here !
Chances are, if you play pokemon, you may have wondered how many different teams you can make in the game. Well wonder no longer, because today I'll be answering that question ! Quick disclaimer: what I'm going to be calculating here is an upper bond for the possible number of teams, as the actual number would be really unpractical to calculate, for several reasons that will become apparent as I progress step by step.

1. The Pokemons​

So, if we wanna calculate the number of possible teams, the first thing we need to know is how many combinations of 6 different pokemon you can make. For the sake of simplicity, we'll be assuming that we're playing the game on cartridge, which allows us to have multiples of the same pokemon on our team. According to bulbapedia, as of USUM there are 907 pokemons including megas and alternate formes. Now, when composing a team of 6 pokemon, the process you actually go through is pick a pokemon amongst those 907 six times in a row, which means that this gives us 907 ^ 6 possibilities. However, a lot of these combinations actually result in the same team, as the order in which you pick these pokemon does not matter (for example, picking Pikachu and then Salamence is the same as picking Salamence then Pikachu). To eliminate duplicates, we'll need to figure out how much teams you can make with the same 6 pokemons, and then divide our first number by that. Following the same principle, making teams from 6 set pokemons comes down to picking one amonst these 6, then one amongst the 5 remaining ones (because this time you can't pick the same one twice), and etc, which gives us 6*5*4*3*2*1 (commonly denoted 6!) possibilities for teams using 6 pokemon. In the end, this means that there are (907 ^ 6)/6! possibilities for different teams of 6 pokemon, which gives us the already humongous number of 7.7323451e+14 possibilities. But wait, there's more !

2. The Movesets​

For each of these 6 pokemons, you'll need to pick 4 moves. Now, it can be quite hard to calculate the number of possible movesets because not every pokemon has a movepool of the same size. To circumvent this problem, we'll calculate the upper bond for that instead. The pokemon that can learn the most moves (excluding Smeargle, obviously) in the game is Mew with a whopping 179 moves to chose from, which means that you can pick from at most 179 moves when making a moveset for each pokemon. You can't pick the same move twice when making a moveset, which means we get to chose one out of 179 for the first move, one out of 178 for the second move, one out of 177 for the third one and finally one out of 176 for the last one, which gives us (at most) 179 * 178 * 177 * 176 = 95872758 possibilities for each pokemon's moveset. Once again, the order in which these are chosen does not matter, so following the same logic as before, we'll have to divide our number by 4!, which brings us at 3994698 possibilities for distinct movesets. Now, since you have to pick a moveset for each of the 6 pokemon in the team, that means you have to pick one moveset out of 3994698 six times, which means that there are 3994698 ^ 6 = 4.0635323e+39 possibilities for moveset configurations in the team.

3. Abilities​

For the sake of simplicity, we'll take an upper bond and consider that every pokemon gets to chose from 3 possible abilities. This gives us 3^6 = 729 possibilities for ability configurations in the team.

4. EVs and IVs
​

Now, each of these pokemon has an EV and IV spread and a nature. IVs are pretty straightforward : for each stat, you have 32 possibilities ranging from 0 to 31 and there are 6 stats, which gives us 32 ^ 6 = 1073741824 different IV spreads for one given pokemon, following the same logic as before (because this time there can be no duplicates : if you pick 27 for hp and 31 for attack, it is NOT the same as if you did the opposite) which we'll have to take to the power of 6 since there are 6 pokemons in our team, bringing us at . EVs are a little bit more complicated. Similar to IVs, it comes down to picking an EV value for each stat, ranging from 0 to 255, which gives us 256 possibilities for each stats. However, this time, you cannot exceed 510 EV points in total. This is where it gets difficult. In order to calculate the number of legal EV spreads, we'll take the total amount of EV spreads and figure out what is the percentage of spreads that do not exceed 510 EVs in total. First, we need to know what is the maximum number of EV points a pokemon could have if it was allowed to have 255 in each stat. That number is 255 * 6 = 1530. Since a legal EV spread can have a total of 510 EV points at most, we can conclude that the percentage of legal EV spreads is 510 / 1530 (what this number actually represents is the probability of getting a number that is less than or equal to 510 when picking a number at random between 1 and 1530), which is exactly 1 / 3 or 33%. So we'll take the total amount of EV spreads, which is 256 ^ 6 (once again following the same logic as before) and multiply it by the percentage of legal EV spreads to get the number of legal EV spreads, which turns out to be 9.3824992e+13 for one given pokemon. Since there are still six pokemons on the team (I'm surprised none of them have ran away at this point because this post is getting really long), we'll once again need to take both of these numbers to the power of 6, and then multiply them with eachother to get the total number of EV and IV configurations for one given team, which is an astronomical 1.045467e+138. Yikes.

5. Nature
​

This one is the most straightforward. There are 25 different natures to pick from, 21 if you consider that all of the 5 natures which do not affect your pokemons' stats to be one and the same (which is true for competitive purposes), so there are 21 ^ 6 = 85766121 possibilities for nature configurations in a team. Moving on.

6. Items
​

Each pokemon can hold an item, and there are about 900 items in the game as of USUM. However for our purposes, we only need to know the number of held items, which I had to manually count in the Showdown teambuilder. There are 379 of these. My eyes hurt T_T. Same as before, you have to pick one for each pokemon which gives us 379 ^ 6 = 2.963707e+15 possibilities for item configurations in a team.

7. The (really) GRAND TOTAL (finally!)
​

Now that we have the number for all possible combinations of pokemon, movesets, EV spreads, IV spreads, natures and items, within a team, all we have to do is take all these numbers and multiply them together !

7.7323451e+14 (Pokemon configurations) * 4.0635323e+39 (Moveset configurations) * 729 (Abilty configurations) * 1.045467e+138 (EV and IV spread configurations) * 85766121 (Nature configurations) * 2.963707e+15 (Item configurations) = 6.08701e+218 possible team configurations in Pokemon.

...Wow. To give you something to compare that to, there are approximatively 10^82 atoms in the observable universe. The number of possible team configurations in Pokemon is over 6.08701e+136 bigger than that (which is also bigger than the number of atoms in the observable universe).

If you think that's big, the number of possible Hackmons team is even larger, since it allows every pokemon to pick from all 728 existing moves and all 233 existing abilities.

Anyways, that's all, hope you enjoyed reading this post and congratulations for making it all the way to the end :D

 

[ItalianBaptist]

And then you get into competitive and the number drops to about 10 ;)

Seriously though, it’s really cool that you went to check this out. It furthers a theory I saw that a major reason for Pokémon’s is that it allows people to express their individuality within the community :)

 

[SadisticMystic]

In step 1, I see 868 for the number of meaningfully distinct forms (omitting cosmetic-only differences like Unown, but counting totems as distinct if for no other purpose than weight-based interactions) that you can legally have outside of battle, and therefore at the start of battle. Note that if you try to include both a Mega Venusaur and a Mega Blastoise on your team, you'll bring them into battle as simply Venusaur or Blastoise, and have to choose a maximum of one to mega evolve over its course; the other will never be able to fulfill the form you envisioned it taking. My 868 number does not include alternate forms for Arceus, Silvally, Genesect, or Giratina, all of which are entirely constrained by their item slots and can be accounted for in that section. I'd also recommend not dividing by 6! and simply assuming that you're playing this team in a format where species clause doesn't exist. Otherwise that division isn't sufficient to account for it, you'd also have to account for the fact that you wouldn't be allowed to include both Rotom-Heat and Rotom-Wash on the same team, despite those being separate entries in the list of 868.

If you're going to treat all neutral natures as fungible (and thus use 21 as the power instead of 25), you might as well do the same for all EV points that don't land on multiples of 4. This allows you to group them not as 510 EVs, but as 127 stat points that result from those EVs. From there you can divide the 127 points into 7 buckets: one for each stat, plus a seventh bucket for unallocated stat points (either from not filling out all 510 EVs or assigning them inefficiently so as to have the non-multiples of 4 add up to more than just the usual 2), and there are 124,397,910,208 such distributions. This is an overestimate, because attempting to put more than half of the 127 points into a single bucket is not a valid assignment unless that bucket is the one for Unallocated, but not a very big one. Assuming that at least a majority of the points (64) wound up in one bucket, dividing the other 63 points among the buckets can be done in 1,078,897,248 ways, and as there are six such buckets that are forbidden from being the majority, we lose about 6 billion of those possibilities and are left with 117,924,526,720 (about 1/800 of the figure you've put forth for this component).

If Showdown's teambuilder lists 379 distinct items, remember that some of these items are not legally obtainable in any gen 7 game. Specifically there are 17 gems, 12 berries, 5 types of Poke Ball, Mail, and the Old Amber. Granted there are other items such as Roto Powers that can legally be held in-game despite the teambuilder not listing them, but at some point you have to invoke the redundancy clause here, which may even prune the list of meaningfully distinct items below that 333 (Helix Fossil and Dome Fossil, for instance, behave identically as held items under all possible battle interactions, including Fling).

Gender is also a thing which can give you up to another factor of 64 depending on how many of the species on your team are not absolutely constrained in this regard, and it has meaningful battle interactions (Attract, Captivate, Cute Charm, Rivalry). Might as well count that too.

 

[Delibird Heart]

long and cool answer

In step 1, the reason I divide by 6! has nothing to do with species clause; I'm not counting teams which would be the same : for example Venusaur/Blastoise/Charizard/Pikachu/Snorlax/Lapras is the same as Lapras/Venusaur/Blastoise/Charizard/Pikachu/Snorlax. Since there 6! different ways for the same team to be rearranged, I have to divide the total number of teams by that to make sure there are no such duplicates.

Other than that all your points seem valid and I'll probably take them into account tomorrow when it's not 3AM lol

Thanks for putting this much effort into your answer :)

 

[pulsar512b]

I'm a dumbass majoring in math, so I'm going to figure out the number for BH. I'll come back tomorrow, and do the calculations.

 

[SadisticMystic]

I'm a dumbass majoring in math, so I'm going to figure out the number for BH. I'll come back tomorrow, and do the calculations.

This time, with innate megas and other inaccessible forms like Ash Greninja and Ultra Necrozma being available from the start of battle (but not Primal Groudon, who's on the banlist), there are 915 meaningfully different forms. Again I've excluded Arceus and Silvally forms (because even in hackmons, these forms are purely cosmetic and function as normal type, unless they have both the proper ability and a working item for that ability), and this time I've also gone ahead and excluded the totems (if you want to include them, just use 926 instead of 915). Without loss of generality, we can assume the team is sorted in order of descending Pokedex number, so the six team slots can be filled out in 916^6/720 - 1 ways. Note the use of 916 instead of 915, because it is legal to bring a team of less than 6, and the 916th possibility represents "empty slot"--it just isn't legal to have all six slots be empty, hence the - 1 at the end.

Gender: In BH, any Pokemon (regardless of its species' normal ratio) can be male, female, or genderless, which produces a factor of 3 per Pokemon on the team. This will usually be 729, but may be less if there are blank slots.

Nature: As before, this is a factor of 21 per non-empty slot on the team. Simple enough.

Level: Yep, you can customize this too. That's a factor of 100 per slot.

Happiness: It's arguable whether this should be counted as meaningful, seeing as it only makes a difference in the presence of the moves Return or Frustration, fewer options than there are to make gender relevant. But if it does count, that's a factor of 256.

IVs/EVs: With no global EV budget in G7 hackmons (a reversal of course from the 6 version, due to the in-game mechanics also changing), the math here gets a lot simpler. In fact, the EVs can almost be thought of as extending the limit for each IV to 94, so instead of treating each stat as a range 0-31 and (again ignoring EVs that don't fall on multiples of 4) a separate range 0-63, which would be a factor of 2048 per stat, the two can be combined into one range of 0-94. However, due to intricacies with the move Hidden Power, it may conceivably make a difference if you have that combined component at 93 out of 94 which breaks down as 31 IV and 248 EV, as opposed to breaking it down as 30 IV and 252 EV. 31/244 vs. 29/252 to get 92 out of 94, meanwhile, behave completely identically. "0 out of 94" forces the IV component to be 0, and "94 out of 94" forces it to be 31 (Showdown still doesn't have support for letting you type a custom number and click the Hyper Training flag), but anything else can exist in two variations, one where the IV component is odd and one where it's even. Thus, instead of 95 or 2048 possibilities here, I see 188 per stat that it makes sense to meaningfully disambiguate. Across all six stats, that's a factor of 44,151,665,987,584...which is actually even smaller than the number from having to take the global EV budget into account (but without generalizing the two components into the same idea).

Moves: There are 728 moves currently coded in game data. After taking out the moves that are banned (CFZs, Chatter, Fissure, Horn Drill, Guillotine, Sheer Cold, Double Team, Minimize) we're left with 668. The first moveslot is special compared to the others, as it's the only one that's not allowed to be empty (even in BH, the teambuilder won't validate a Pokemon that has no moves), and it's also the specific move that will be looked at in case one of the user's moves is Conversion and that gets used in battle. The other three moves are completely fungible in all aspects, and are allowed to be empty; assume they are sorted in order of descending ID. This gives us 668*(669^3/6), or 33,335,238,402 ways to fill out a Pokemon's moveset. By adjusting the number of PP Ups on each move, you could try to squeeze another factor of 256 in here, but Showdown has no support for such a thing: it forces Trump Card to take 0 PP Ups at all times, and any other move to take maximum possible PP Ups.

Abilities: There are 233 abilities in the game thus far, 14 of which are banned in BH and another 18 of which are exact clones of other abilities for in-battle purposes (to include abilities that have no purpose in singles, like Plus or Minus, being treated as clones of Illuminate). There's also a blank ability that's coded into the game but not normally obtainable, and only differs from Illuminate here in the case where an opponent tries to Trace it and you switch out to your own Tracer, or else to someone who can steal their Trace with Role Play or Skill Swap. Whether you think that's enough to count as meaningfully distinct or not, that makes a factor of either 201 or 202 per slot. This is a slight overestimate because picking Comatose as the ability puts a constraint on the moveslots (can't have Sleep Talk in that case), and because of the ability clause that prevents the same ability (including clones) from appearing three or more times on the same team, but I'll treat these effects as negligible.

Items: If there are 379 distinct items in the teambuilder (plus the possibility of running itemless), this time we don't have to worry about which ones might be obtainable or not. However, Gengarite is banned, and we have several redundant items as well: the 27 Poke Ball types, the 11 Fossils, and such. I'll figure this around 330 again.

If we simplify and assume there aren't going to be any vacant slots on the team, I get about 1.45e+38 options per Pokemon, or 1.29e+226 options for a Hackmons team as a whole (including the ability to count the full gamut of happiness values). Larger than the number cited above for an AG team, but not by much, and the AG bounds may be subject to further refinement.

 

[pulsar512b]

This time, with innate megas and other inaccessible forms like Ash Greninja and Ultra Necrozma being available from the start of battle (but not Primal Groudon, who's on the banlist), there are 915 meaningfully different forms. Again I've excluded Arceus and Silvally forms (because even in hackmons, these forms are purely cosmetic and function as normal type, unless they have both the proper ability and a working item for that ability), and this time I've also gone ahead and excluded the totems (if you want to include them, just use 926 instead of 915). Without loss of generality, we can assume the team is sorted in order of descending Pokedex number, so the six team slots can be filled out in 916^6/720 - 1 ways. Note the use of 916 instead of 915, because it is legal to bring a team of less than 6, and the 916th possibility represents "empty slot"--it just isn't legal to have all six slots be empty, hence the - 1 at the end.

Gender: In BH, any Pokemon (regardless of its species' normal ratio) can be male, female, or genderless, which produces a factor of 3 per Pokemon on the team. This will usually be 729, but may be less if there are blank slots.

Nature: As before, this is a factor of 21 per non-empty slot on the team. Simple enough.

Level: Yep, you can customize this too. That's a factor of 100 per slot.

Happiness: It's arguable whether this should be counted as meaningful, seeing as it only makes a difference in the presence of the moves Return or Frustration, fewer options than there are to make gender relevant. But if it does count, that's a factor of 256.

IVs/EVs: With no global EV budget in G7 hackmons (a reversal of course from the 6 version, due to the in-game mechanics also changing), the math here gets a lot simpler. In fact, the EVs can almost be thought of as extending the limit for each IV to 94, so instead of treating each stat as a range 0-31 and (again ignoring EVs that don't fall on multiples of 4) a separate range 0-63, which would be a factor of 2048 per stat, the two can be combined into one range of 0-94. However, due to intricacies with the move Hidden Power, it may conceivably make a difference if you have that combined component at 93 out of 94 which breaks down as 31 IV and 248 EV, as opposed to breaking it down as 30 IV and 252 EV. 31/244 vs. 29/252 to get 92 out of 94, meanwhile, behave completely identically. "0 out of 94" forces the IV component to be 0, and "94 out of 94" forces it to be 31 (Showdown still doesn't have support for letting you type a custom number and click the Hyper Training flag), but anything else can exist in two variations, one where the IV component is odd and one where it's even. Thus, instead of 95 or 2048 possibilities here, I see 188 per stat that it makes sense to meaningfully disambiguate. Across all six stats, that's a factor of 44,151,665,987,584...which is actually even smaller than the number from having to take the global EV budget into account (but without generalizing the two components into the same idea).

Moves: There are 728 moves currently coded in game data. After taking out the moves that are banned (CFZs, Chatter, Fissure, Horn Drill, Guillotine, Sheer Cold, Double Team, Minimize) we're left with 668. The first moveslot is special compared to the others, as it's the only one that's not allowed to be empty (even in BH, the teambuilder won't validate a Pokemon that has no moves), and it's also the specific move that will be looked at in case one of the user's moves is Conversion and that gets used in battle. The other three moves are completely fungible in all aspects, and are allowed to be empty; assume they are sorted in order of descending ID. This gives us 668*(669^3/6), or 33,335,238,402 ways to fill out a Pokemon's moveset. By adjusting the number of PP Ups on each move, you could try to squeeze another factor of 256 in here, but Showdown has no support for such a thing: it forces Trump Card to take 0 PP Ups at all times, and any other move to take maximum possible PP Ups.

Abilities: There are 233 abilities in the game thus far, 14 of which are banned in BH and another 18 of which are exact clones of other abilities for in-battle purposes (to include abilities that have no purpose in singles, like Plus or Minus, being treated as clones of Illuminate). There's also a blank ability that's coded into the game but not normally obtainable, and only differs from Illuminate here in the case where an opponent tries to Trace it and you switch out to your own Tracer, or else to someone who can steal their Trace with Role Play or Skill Swap. Whether you think that's enough to count as meaningfully distinct or not, that makes a factor of either 201 or 202 per slot. This is a slight overestimate because picking Comatose as the ability puts a constraint on the moveslots (can't have Sleep Talk in that case), and because of the ability clause that prevents the same ability (including clones) from appearing three or more times on the same team, but I'll treat these effects as negligible.

Items: If there are 379 distinct items in the teambuilder (plus the possibility of running itemless), this time we don't have to worry about which ones might be obtainable or not. However, Gengarite is banned, and we have several redundant items as well: the 27 Poke Ball types, the 11 Fossils, and such. I'll figure this around 330 again.

If we simplify and assume there aren't going to be any vacant slots on the team, I get about 1.45e+38 options per Pokemon, or 1.29e+226 options for a Hackmons team as a whole (including the ability to count the full gamut of happiness values). Larger than the number cited above for an AG team, but not by much, and the AG bounds may be subject to further refinement.

Thanks for the snipe. I'll probably do the math for pure hackmons, and maybe even in some other metagames for fun lol. DO NOT SNIPE ME.

 

[pulsar512b]

Nice post but one thing you forgot about is moveset incompatibility like for example No guard + Fissure machamp is illegal. I know I am nitpicking somewhat and I would be hard to calculate for every single moveset incompatibility but I just wanted to mention it

IE impossible. And anyways, this is only a rough estimate.

 

[Pikachu315111]

Think would have just been easier to just use this image to explain the number of possibilities :

 

[BP]

I'm actually really curious to find out the exact number of teams possible that you could possibly have. Even with event mons and what not. This is a really cool idea and I hope people keep working on this.

 

[Zowayix]

IVs/EVs: With no global EV budget in G7 hackmons (a reversal of course from the 6 version, due to the in-game mechanics also changing), the math here gets a lot simpler.

Wait, EV mechanics are different between Gens 5 - 7? Hearing that hackmons' EV mechanics changed from Gen 5 to 6 is surprising enough; hearing that it changed again from Gen 6 to Gen 7 is even more of a surprise.

Edit: Interesting, thanks!

 

[SadisticMystic]

The core EV mechanics have been the same ever since the modern system was adopted (with the exception that starting in XY, it prunes the "useless" points beyond 252 from prior-gen transfers, and doesn't let you accumulate them in the first place for Pokemon that originated there). What changed about 6 is that for that generation only, the piece of legality-checking that rejected a Pokemon for having too many EVs was replicated client-side, so that you couldn't dodge it even in local battles where the players weren't connecting to the PGL server. When 7 came along, they changed it back so the only check was server-side, much like the checks for more visible problems like illegal moves, and the BH leadership team decided to faithfully implement the respective levels of checking in each generation's case.

 

[Champion Leon]

I feel like posts such as yours deserve an award in the Nintendo Hall Of Fame for a mathematician’s level of calculations and direct accuracy.

Too bad Gen 8 forces a re-configuration, with all of the new additions including Gigatamax forms (not just Dynamax).

Someone should honestly commend anyone on these sorts of posts.

If I were to grade you I would give you an S+. As even an A+ wouldn’t be enough as it is beyond perfect.

This time, with innate megas and other inaccessible forms like Ash Greninja and Ultra Necrozma being available from the start of battle (but not Primal Groudon, who's on the banlist), there are 915 meaningfully different forms. Again I've excluded Arceus and Silvally forms (because even in hackmons, these forms are purely cosmetic and function as normal type, unless they have both the proper ability and a working item for that ability), and this time I've also gone ahead and excluded the totems (if you want to include them, just use 926 instead of 915). Without loss of generality, we can assume the team is sorted in order of descending Pokedex number, so the six team slots can be filled out in 916^6/720 - 1 ways. Note the use of 916 instead of 915, because it is legal to bring a team of less than 6, and the 916th possibility represents "empty slot"--it just isn't legal to have all six slots be empty, hence the - 1 at the end.

Gender: In BH, any Pokemon (regardless of its species' normal ratio) can be male, female, or genderless, which produces a factor of 3 per Pokemon on the team. This will usually be 729, but may be less if there are blank slots.

Nature: As before, this is a factor of 21 per non-empty slot on the team. Simple enough.

Level: Yep, you can customize this too. That's a factor of 100 per slot.

Happiness: It's arguable whether this should be counted as meaningful, seeing as it only makes a difference in the presence of the moves Return or Frustration, fewer options than there are to make gender relevant. But if it does count, that's a factor of 256.

IVs/EVs: With no global EV budget in G7 hackmons (a reversal of course from the 6 version, due to the in-game mechanics also changing), the math here gets a lot simpler. In fact, the EVs can almost be thought of as extending the limit for each IV to 94, so instead of treating each stat as a range 0-31 and (again ignoring EVs that don't fall on multiples of 4) a separate range 0-63, which would be a factor of 2048 per stat, the two can be combined into one range of 0-94. However, due to intricacies with the move Hidden Power, it may conceivably make a difference if you have that combined component at 93 out of 94 which breaks down as 31 IV and 248 EV, as opposed to breaking it down as 30 IV and 252 EV. 31/244 vs. 29/252 to get 92 out of 94, meanwhile, behave completely identically. "0 out of 94" forces the IV component to be 0, and "94 out of 94" forces it to be 31 (Showdown still doesn't have support for letting you type a custom number and click the Hyper Training flag), but anything else can exist in two variations, one where the IV component is odd and one where it's even. Thus, instead of 95 or 2048 possibilities here, I see 188 per stat that it makes sense to meaningfully disambiguate. Across all six stats, that's a factor of 44,151,665,987,584...which is actually even smaller than the number from having to take the global EV budget into account (but without generalizing the two components into the same idea).

Moves: There are 728 moves currently coded in game data. After taking out the moves that are banned (CFZs, Chatter, Fissure, Horn Drill, Guillotine, Sheer Cold, Double Team, Minimize) we're left with 668. The first moveslot is special compared to the others, as it's the only one that's not allowed to be empty (even in BH, the teambuilder won't validate a Pokemon that has no moves), and it's also the specific move that will be looked at in case one of the user's moves is Conversion and that gets used in battle. The other three moves are completely fungible in all aspects, and are allowed to be empty; assume they are sorted in order of descending ID. This gives us 668*(669^3/6), or 33,335,238,402 ways to fill out a Pokemon's moveset. By adjusting the number of PP Ups on each move, you could try to squeeze another factor of 256 in here, but Showdown has no support for such a thing: it forces Trump Card to take 0 PP Ups at all times, and any other move to take maximum possible PP Ups.

Abilities: There are 233 abilities in the game thus far, 14 of which are banned in BH and another 18 of which are exact clones of other abilities for in-battle purposes (to include abilities that have no purpose in singles, like Plus or Minus, being treated as clones of Illuminate). There's also a blank ability that's coded into the game but not normally obtainable, and only differs from Illuminate here in the case where an opponent tries to Trace it and you switch out to your own Tracer, or else to someone who can steal their Trace with Role Play or Skill Swap. Whether you think that's enough to count as meaningfully distinct or not, that makes a factor of either 201 or 202 per slot. This is a slight overestimate because picking Comatose as the ability puts a constraint on the moveslots (can't have Sleep Talk in that case), and because of the ability clause that prevents the same ability (including clones) from appearing three or more times on the same team, but I'll treat these effects as negligible.

Items: If there are 379 distinct items in the teambuilder (plus the possibility of running itemless), this time we don't have to worry about which ones might be obtainable or not. However, Gengarite is banned, and we have several redundant items as well: the 27 Poke Ball types, the 11 Fossils, and such. I'll figure this around 330 again.

If we simplify and assume there aren't going to be any vacant slots on the team, I get about 1.45e+38 options per Pokemon, or 1.29e+226 options for a Hackmons team as a whole (including the ability to count the full gamut of happiness values). Larger than the number cited above for an AG team, but not by much, and the AG bounds may be subject to further refinement.

In step 1, the reason I divide by 6! has nothing to do with species clause; I'm not counting teams which would be the same : for example Venusaur/Blastoise/Charizard/Pikachu/Snorlax/Lapras is the same as Lapras/Venusaur/Blastoise/Charizard/Pikachu/Snorlax. Since there 6! different ways for the same team to be rearranged, I have to divide the total number of teams by that to make sure there are no such duplicates.

Other than that all your points seem valid and I'll probably take them into account tomorrow when it's not 3AM lol

Thanks for putting this much effort into your answer :)

 

[BBlaine103]

Long winded math. My favorite. Thank you for this. I was wondering and need the math for a paper thx. Hippity hoppity your math is my property.