The Olympian
Banned deucer.
Hi all, in lieu of generation 8, I thought it would be cool to answer a question that many have sought the solution to: How many possible Pokemon teams are there? I've encountered several threads on Reddit and a few on here that have attempted to answer this question, but I've never seen someone provide an answer that does not leave out important details. I will attempt to provide the true answer, but bear with me because the numbers that it ensues are massive, and I mean, MASSIVE. To provide buildup for the practically infinite number of possible teams in generation 7, I will begin with generation 1 and work my way there.
*note: The answers provided for each generation will follow the "Anything Goes" format, which is the equivalent of "Singles - No Restrictions" in the Pokemon games.
Generation 1
In the days of RBY, Pokemon battling was a lot simpler. There were no items or abilities, no natures, and no alternate forms for each Pokemon. There were 151 Pokemon and each one could be leveled from 1-100 (although some evolved forms could not be lower than certain levels), and there were a total of 165 moves, with move-pools ranging in size from 1 for Ditto to 57 for Mew. Calculating the number of possible teams based on Pokemon species alone is quite simple; it can be represented by the expression (151^6), since there are 151 Pokemon and a team consists of 6 of them. However, this only accounts for the possible combinations of 6 Pokemon, and leaves out team combinations of 5, 4, 3, 2, and 1. I will be leaving out these combinations for my calculation, because they are not relevant to the number of useful teams. To further simplify my initial calculation to represent the number of useful teams, I will only include level 100 Pokemon, since in generation 1 there is no incentive to battle with lower-leveled creatures. I will also be leaving out the possible EV and IV combinations, since there is no incentive to not have maximum EV and IV investment in each base stat. An important thing to note about the 151^6 expression is that it assumes that the order of the Pokemon on the team does matter, which in the case of generation 1 is partially true. Since the Pokemon that is in the first slot of the party is sent out first, the Pokemon that occupies this slot on the team does matter, and the order of the remaining 5 does not. In a scenario where the order of all 6 Pokemon doesn't matter, but repeats of Pokemon are allowed, the number of possible teams would be equal to (156!) / (6! * 150!). To account for the one slot of the 6 that does matter, we can take the scenario of (151 choose 5 repeats allowed) and multiply it by 151. And now, the first key part of this calculation is shown below:
Number of possible useful teams: 151 * [ (156!) / (5! * 151!) ] = 108,970,197,336 ~ Approximately 109 Billion
After retrieving this number my next inclination was to calculate the number of possible move-pool combinations, but it then occurred to me that in terms of simply calculating the number of possible useful teams, taking into account move-pools is actually not necessary. Why, you may ask? Well, in the large scope of things the actual number of teams that could be considered useful is quite subjective. Theoretically, in every generation there may exist a single Pokemon team combination that is more consistent than any other possible team, and this goes for any Smogon meta game as well (not just Singles - No Restrictions). The problem though is that it is very likely that not a single one of these teams has ever been made by human hands, because it would take a sophisticated computer algorithm to actually create such a team (in later generations, such as generation 7, the insane complexity of such a calculation would probably be bordering on impossible). Because of this, I define a useful team as a fully optimized team in terms of the Pokemon that are present on it (in other words, there are many possible variations of 6 Mewtwo teams in generation 1, but only 1 of them is the best built one). Therefore, the number that I have listed in bold above is the exact number of possible fully optimized teams in which all Pokemon species are used. I will now provide this same calculation for generations 2 - 7, and afterwards I will provide an answer to the total number of possible RBY Pokemon teams to showcase just how ridiculously massive the numbers can get.
Generation 2
Completing the calculation for GSC is actually made easy due to it following the same exact formula as generation 1. In generation 2 there are 251 Pokemon, each one lacking alternate forms, and so our equation is as follows:
Number of possible useful teams: 251 * [ (256!) / (5! * 251!) ] = 2,211,196,813,056 or ~ 2.211 Trillion
Generation 3
The calculation for the ADV meta game is also simplified by earlier calculations. There are 386 Pokemon in this generation, but Deoxys has 4 forms that are distinct from each other, bringing the actual number to 389.
Number of possible useful teams: 389 * [ (394!) / (5! * 389!) ] = 30,004,373,218,092 or ~ 30 Trillion
Generation 4
The DPP generation has 493 Pokemon, however the following Pokemon have distinct alternate forms: Arceus (17), Deoxys (4), Giratina (2), Shaymin (2), Rotom (6), & Wormadam (2). This brings our total number of Pokemon to 520 (it should be noted that there is also an alternate form of Pichu, however this form does not differ at all competitively from regular Pichu).
Number of possible useful teams: 520 * [ (525!) / (5! * 520!) ] = 169,559,685,804,600 or ~ 169.6 Trillion
Generation 5
The BW/BW2 Pokedex contains 649 entries, with the following Pokemon having alternate forms: Arceus (17), Deoxys (4), Genesect (5), Giratina (2), Shaymin (2), Tornadus (2), Thundurus (2), Landorus (2), Rotom (6), & Wormadam (2), which brings our total to 683 (Keldeo and Basculin have alternate forms that do not differ competitively from their normal forms). The interesting case about generation 5 however, is Kyurem. Kyurem has 3 unique forms, but both Black Kyurem and White Kyurem can't be used more than once on a team. Additionally, generation 5 is the first generation in which the order of all 6 Pokemon on the team does not matter, since the Pokemon in the first slot is not forced to be the lead. This creates some slight tweaks to the formula that had been used in the previous calculations:
Number of possible useful teams: (689!) / (6! * 683!) + [ (689!) / (6! * 683!) - (688!) / (6! * 682!) ] = 146,645,341,878,840 or ~ 146.6 Trillion
*note: Notice how the number for this generation is actually smaller than the previous one, due to the significant impact created by the Pokemon occupying the first slot in the party not mattering. In the previous formulas I had taken the number of combinations of 5 Pokemon (with repeats allowed) and multiplied it by the total number of Pokemon, but in this formula I instead took the number of combinations of 6 Pokemon right off the bat. To correctly account for the Kyurem forms I brought the total number of Pokemon from 683 to 684, and then added the difference between (684 choose 6 repeats allowed) and (683 choose 6 repeats allowed), since Black Kyurem can be swapped with White Kyurem. Now you may be wondering, why would I give the Kyurem forms the "repeat" treatment if they can't be repeated. Well, in this instance it is due to a matter of mathematical equivalency. (684 choose 6 repeats allowed) is mathematically equivalent to (689 choose 6 repeats not allowed), so there is no error.
Generation 6
This is where things begin to get more complicated, with the introduction of Mega forms. Our basic number of Pokemon is already at 721, and Pokemon with alternate forms that affect them competitively (excluding Megas) are Arceus (18), Deoxys (4), Genesect (5), Giratina (2), Groudon (2), Hoopa (2), Kyogre (2), Shaymin (2), Landorus (2), Thundurus (2), Tornadus (2), Rotom (6), Gourgeist (4), Pumpkaboo (4), Meowstic (2), & Wormadam (2), and Pikachu (2), bringing our total to 767 (the variants of Pikachu are counted as 1 alternate form since they do not differ competitively from each other). From here, we also have to account for Black Kyurem and White Kyurem, as well as the 48 mega forms. Each mega form will be treated as a separate Pokemon that can't be repeated, and the X & Y mega forms of Mewtwo and Charizard will be given the same treatment as the Kyurem forms.
Number of possible useful teams: (819!) / (6! * 813!) + [ (819!) / (6! * 813!) - (816! / (6! * 810!) ] = 420,516,939,664,976 or ~ 420.5 Trillion
*note: For this calculation the total number of Pokemon is actually treated as 814 (767 Pokemon + 1 for Black/White Kyurem + 46 Mega Pokemon). The difference between (819 choose 6 no repeats) and (816 choose 6 no repeats) is then added to the result to account for the Kyurems and X/Y Megas.
Generation 7
At last, gen 7. We have 809 entries in this monstrous Pokedex, with the same number of Megas. Meltan and Melmetal do not count, however, so we have to subtract 2. Pokemon with significant alternate forms include Arceus (18), Deoxys (4), Genesect (5), Giratina (2), Landorus (2), Thundurus (2), Tornadus (2), Shaymin (2), Kyogre (2), Hoopa (2), Groudon (2), Rotom (6), Gourgeist (4), Pumpkaboo (4), Meowstic (2), Wormadam (2), Zygarde (4), Oricorio (4), Silvally (18), Lycanroc (3), Greninja (2), Pikachu (2), 19 Alolan forms, & 11 Totem forms. This brings our grand total of Pokemon (excluding Megas and Kyurem-effect forms) to 909. Dusk Mane and Dawn Wings Necrozma, as well as Ultra Necrozma, have the same effect on our formula as Mega forms.
Number of possible useful teams: (965!) / (6! * 959!) + [ (965!) / (6! * 959!) - (962!) / (6! * 956!) ] = 1,124,741,819,064,656: 1.125 Quadrillion
*note: The total number of Pokemon here is treated as 960 (909 Pokemon + 1 for Black/White Kyurem + 46 Mega Pokemon + 4 for the Necrozma forms, Ultra Necrozma is the equivalent of adding 2 Megas).
Links To Assist Those Who Want a Better Idea of How These Formulas Work
What is the Importance of These Numbers?
While to some this thread may come across as unimportant and a waste of its maker's time, the numbers that have been produced here are actually more important than they may seem. Each number can be thought of as a "population size" that denotes the total number of competitive teams that can be made in each generation's "AG" format. The same approaches used in the calculations that I have provided can be used to denote the competitive team population size of any Smogon/Pokemon meta game. Why though, does the pool size of competitive teams even matter? In the game of Pokemon, there are 2 main aspects that affect the outcome of a match/skill of a player. 1.) The team that they are using, and 2.) the player's level of ability to play the game. Assuming that both of these aspects follow a normal distribution, it is technically possible to calculate a theoretical "max ladder rating (whether it be Elo or Glicko/GXE)" for any meta game. The complications of that rabbit hole, however, can be saved for another day.
What Would These Numbers Look Like if They Accounted for All Possible Team Combinations Rather Than Only "Useful/Competitive" Ones?
To put it short and sweet, they would be ridiculously massive. I was originally going to calculate these numbers for every generation, but they are so huge that calculating them would be tedious to a point in which it would take weeks to calculate all 7 of them. However, as promised I will provide the answer for the RBY gen only (and unfortunately, an estimated answer at that). Each generation after would produce a significantly larger number than the previous generation. And honestly, these numbers would have no intrinsic value other than being cool because of their size.
For RBY, we have to use the original (151)^6 number, and then add (151)^5, (151)^4, (151)^3, (151)^2, and 151 to it. The resulting number is 11,932,937,665,656 which is already larger than the original number derived for generation 2. This represents the total number of permutations of all generation 1 Pokemon, with teams of 6,5,4,3,2 and 1 Pokemon all being accounted for. Next we would have to multiply the result by 100, since every Pokemon can be leveled 1-100 (there are exceptions to this, in fact many, but we are doing this for simplicity's sake). Next we would have to account for move-pool size and EV / IV variations. Assuming a median move-pool size of 29 (in order to actually derive the exact answer here, we would have to view the move-pool size of every individual Pokemon and do a separate combination calculation for each), and each Pokemon getting 4 moves, the number of move-pool variations would be ~ 707,281 (this actually leaves out instances in which there are empty move slots). Since there are 252 EVs in each stat and 15 DVs (in generation 1 IVs are referred to as DVs), the total number of EV/DV combinations for each Pokemon is equal to (253^6)*(16^6), which comes to... A tremendously large number (4.3999ee21). Multiplying this number by 707,281, then further by 100, and then further by 11,932,937,665,656 yields approximately ~ 3.7ee42 (37 with 41 zeroes after it). If we wanted to complicate this calculation to the absolute max, we could account for the possible nickname combinations for every Pokemon, and lastly the status condition of each Pokemon (since in generation 1 it is possible to have your Pokemon poisoned/paralyzed etc. before a link battle). The number that this would produce would be mind boggling, easily greater than 1 X 10^100. Generation 7 could potentially produce a number greater than 1 X 10^1000, I wouldn't be surprised if it was a lot bigger than that actually.
This concludes my "presentation." If you have any questions or comments, or if you caught a mistake that I made that I missed, feel free to reply with your thoughts.
*note: The answers provided for each generation will follow the "Anything Goes" format, which is the equivalent of "Singles - No Restrictions" in the Pokemon games.
Generation 1
In the days of RBY, Pokemon battling was a lot simpler. There were no items or abilities, no natures, and no alternate forms for each Pokemon. There were 151 Pokemon and each one could be leveled from 1-100 (although some evolved forms could not be lower than certain levels), and there were a total of 165 moves, with move-pools ranging in size from 1 for Ditto to 57 for Mew. Calculating the number of possible teams based on Pokemon species alone is quite simple; it can be represented by the expression (151^6), since there are 151 Pokemon and a team consists of 6 of them. However, this only accounts for the possible combinations of 6 Pokemon, and leaves out team combinations of 5, 4, 3, 2, and 1. I will be leaving out these combinations for my calculation, because they are not relevant to the number of useful teams. To further simplify my initial calculation to represent the number of useful teams, I will only include level 100 Pokemon, since in generation 1 there is no incentive to battle with lower-leveled creatures. I will also be leaving out the possible EV and IV combinations, since there is no incentive to not have maximum EV and IV investment in each base stat. An important thing to note about the 151^6 expression is that it assumes that the order of the Pokemon on the team does matter, which in the case of generation 1 is partially true. Since the Pokemon that is in the first slot of the party is sent out first, the Pokemon that occupies this slot on the team does matter, and the order of the remaining 5 does not. In a scenario where the order of all 6 Pokemon doesn't matter, but repeats of Pokemon are allowed, the number of possible teams would be equal to (156!) / (6! * 150!). To account for the one slot of the 6 that does matter, we can take the scenario of (151 choose 5 repeats allowed) and multiply it by 151. And now, the first key part of this calculation is shown below:
Number of possible useful teams: 151 * [ (156!) / (5! * 151!) ] = 108,970,197,336 ~ Approximately 109 Billion
After retrieving this number my next inclination was to calculate the number of possible move-pool combinations, but it then occurred to me that in terms of simply calculating the number of possible useful teams, taking into account move-pools is actually not necessary. Why, you may ask? Well, in the large scope of things the actual number of teams that could be considered useful is quite subjective. Theoretically, in every generation there may exist a single Pokemon team combination that is more consistent than any other possible team, and this goes for any Smogon meta game as well (not just Singles - No Restrictions). The problem though is that it is very likely that not a single one of these teams has ever been made by human hands, because it would take a sophisticated computer algorithm to actually create such a team (in later generations, such as generation 7, the insane complexity of such a calculation would probably be bordering on impossible). Because of this, I define a useful team as a fully optimized team in terms of the Pokemon that are present on it (in other words, there are many possible variations of 6 Mewtwo teams in generation 1, but only 1 of them is the best built one). Therefore, the number that I have listed in bold above is the exact number of possible fully optimized teams in which all Pokemon species are used. I will now provide this same calculation for generations 2 - 7, and afterwards I will provide an answer to the total number of possible RBY Pokemon teams to showcase just how ridiculously massive the numbers can get.
Generation 2
Completing the calculation for GSC is actually made easy due to it following the same exact formula as generation 1. In generation 2 there are 251 Pokemon, each one lacking alternate forms, and so our equation is as follows:
Number of possible useful teams: 251 * [ (256!) / (5! * 251!) ] = 2,211,196,813,056 or ~ 2.211 Trillion
Generation 3
The calculation for the ADV meta game is also simplified by earlier calculations. There are 386 Pokemon in this generation, but Deoxys has 4 forms that are distinct from each other, bringing the actual number to 389.
Number of possible useful teams: 389 * [ (394!) / (5! * 389!) ] = 30,004,373,218,092 or ~ 30 Trillion
Generation 4
The DPP generation has 493 Pokemon, however the following Pokemon have distinct alternate forms: Arceus (17), Deoxys (4), Giratina (2), Shaymin (2), Rotom (6), & Wormadam (2). This brings our total number of Pokemon to 520 (it should be noted that there is also an alternate form of Pichu, however this form does not differ at all competitively from regular Pichu).
Number of possible useful teams: 520 * [ (525!) / (5! * 520!) ] = 169,559,685,804,600 or ~ 169.6 Trillion
Generation 5
The BW/BW2 Pokedex contains 649 entries, with the following Pokemon having alternate forms: Arceus (17), Deoxys (4), Genesect (5), Giratina (2), Shaymin (2), Tornadus (2), Thundurus (2), Landorus (2), Rotom (6), & Wormadam (2), which brings our total to 683 (Keldeo and Basculin have alternate forms that do not differ competitively from their normal forms). The interesting case about generation 5 however, is Kyurem. Kyurem has 3 unique forms, but both Black Kyurem and White Kyurem can't be used more than once on a team. Additionally, generation 5 is the first generation in which the order of all 6 Pokemon on the team does not matter, since the Pokemon in the first slot is not forced to be the lead. This creates some slight tweaks to the formula that had been used in the previous calculations:
Number of possible useful teams: (689!) / (6! * 683!) + [ (689!) / (6! * 683!) - (688!) / (6! * 682!) ] = 146,645,341,878,840 or ~ 146.6 Trillion
*note: Notice how the number for this generation is actually smaller than the previous one, due to the significant impact created by the Pokemon occupying the first slot in the party not mattering. In the previous formulas I had taken the number of combinations of 5 Pokemon (with repeats allowed) and multiplied it by the total number of Pokemon, but in this formula I instead took the number of combinations of 6 Pokemon right off the bat. To correctly account for the Kyurem forms I brought the total number of Pokemon from 683 to 684, and then added the difference between (684 choose 6 repeats allowed) and (683 choose 6 repeats allowed), since Black Kyurem can be swapped with White Kyurem. Now you may be wondering, why would I give the Kyurem forms the "repeat" treatment if they can't be repeated. Well, in this instance it is due to a matter of mathematical equivalency. (684 choose 6 repeats allowed) is mathematically equivalent to (689 choose 6 repeats not allowed), so there is no error.
Generation 6
This is where things begin to get more complicated, with the introduction of Mega forms. Our basic number of Pokemon is already at 721, and Pokemon with alternate forms that affect them competitively (excluding Megas) are Arceus (18), Deoxys (4), Genesect (5), Giratina (2), Groudon (2), Hoopa (2), Kyogre (2), Shaymin (2), Landorus (2), Thundurus (2), Tornadus (2), Rotom (6), Gourgeist (4), Pumpkaboo (4), Meowstic (2), & Wormadam (2), and Pikachu (2), bringing our total to 767 (the variants of Pikachu are counted as 1 alternate form since they do not differ competitively from each other). From here, we also have to account for Black Kyurem and White Kyurem, as well as the 48 mega forms. Each mega form will be treated as a separate Pokemon that can't be repeated, and the X & Y mega forms of Mewtwo and Charizard will be given the same treatment as the Kyurem forms.
Number of possible useful teams: (819!) / (6! * 813!) + [ (819!) / (6! * 813!) - (816! / (6! * 810!) ] = 420,516,939,664,976 or ~ 420.5 Trillion
*note: For this calculation the total number of Pokemon is actually treated as 814 (767 Pokemon + 1 for Black/White Kyurem + 46 Mega Pokemon). The difference between (819 choose 6 no repeats) and (816 choose 6 no repeats) is then added to the result to account for the Kyurems and X/Y Megas.
Generation 7
At last, gen 7. We have 809 entries in this monstrous Pokedex, with the same number of Megas. Meltan and Melmetal do not count, however, so we have to subtract 2. Pokemon with significant alternate forms include Arceus (18), Deoxys (4), Genesect (5), Giratina (2), Landorus (2), Thundurus (2), Tornadus (2), Shaymin (2), Kyogre (2), Hoopa (2), Groudon (2), Rotom (6), Gourgeist (4), Pumpkaboo (4), Meowstic (2), Wormadam (2), Zygarde (4), Oricorio (4), Silvally (18), Lycanroc (3), Greninja (2), Pikachu (2), 19 Alolan forms, & 11 Totem forms. This brings our grand total of Pokemon (excluding Megas and Kyurem-effect forms) to 909. Dusk Mane and Dawn Wings Necrozma, as well as Ultra Necrozma, have the same effect on our formula as Mega forms.
Number of possible useful teams: (965!) / (6! * 959!) + [ (965!) / (6! * 959!) - (962!) / (6! * 956!) ] = 1,124,741,819,064,656: 1.125 Quadrillion
*note: The total number of Pokemon here is treated as 960 (909 Pokemon + 1 for Black/White Kyurem + 46 Mega Pokemon + 4 for the Necrozma forms, Ultra Necrozma is the equivalent of adding 2 Megas).
Links To Assist Those Who Want a Better Idea of How These Formulas Work
What is the Importance of These Numbers?
While to some this thread may come across as unimportant and a waste of its maker's time, the numbers that have been produced here are actually more important than they may seem. Each number can be thought of as a "population size" that denotes the total number of competitive teams that can be made in each generation's "AG" format. The same approaches used in the calculations that I have provided can be used to denote the competitive team population size of any Smogon/Pokemon meta game. Why though, does the pool size of competitive teams even matter? In the game of Pokemon, there are 2 main aspects that affect the outcome of a match/skill of a player. 1.) The team that they are using, and 2.) the player's level of ability to play the game. Assuming that both of these aspects follow a normal distribution, it is technically possible to calculate a theoretical "max ladder rating (whether it be Elo or Glicko/GXE)" for any meta game. The complications of that rabbit hole, however, can be saved for another day.
What Would These Numbers Look Like if They Accounted for All Possible Team Combinations Rather Than Only "Useful/Competitive" Ones?
To put it short and sweet, they would be ridiculously massive. I was originally going to calculate these numbers for every generation, but they are so huge that calculating them would be tedious to a point in which it would take weeks to calculate all 7 of them. However, as promised I will provide the answer for the RBY gen only (and unfortunately, an estimated answer at that). Each generation after would produce a significantly larger number than the previous generation. And honestly, these numbers would have no intrinsic value other than being cool because of their size.
For RBY, we have to use the original (151)^6 number, and then add (151)^5, (151)^4, (151)^3, (151)^2, and 151 to it. The resulting number is 11,932,937,665,656 which is already larger than the original number derived for generation 2. This represents the total number of permutations of all generation 1 Pokemon, with teams of 6,5,4,3,2 and 1 Pokemon all being accounted for. Next we would have to multiply the result by 100, since every Pokemon can be leveled 1-100 (there are exceptions to this, in fact many, but we are doing this for simplicity's sake). Next we would have to account for move-pool size and EV / IV variations. Assuming a median move-pool size of 29 (in order to actually derive the exact answer here, we would have to view the move-pool size of every individual Pokemon and do a separate combination calculation for each), and each Pokemon getting 4 moves, the number of move-pool variations would be ~ 707,281 (this actually leaves out instances in which there are empty move slots). Since there are 252 EVs in each stat and 15 DVs (in generation 1 IVs are referred to as DVs), the total number of EV/DV combinations for each Pokemon is equal to (253^6)*(16^6), which comes to... A tremendously large number (4.3999ee21). Multiplying this number by 707,281, then further by 100, and then further by 11,932,937,665,656 yields approximately ~ 3.7ee42 (37 with 41 zeroes after it). If we wanted to complicate this calculation to the absolute max, we could account for the possible nickname combinations for every Pokemon, and lastly the status condition of each Pokemon (since in generation 1 it is possible to have your Pokemon poisoned/paralyzed etc. before a link battle). The number that this would produce would be mind boggling, easily greater than 1 X 10^100. Generation 7 could potentially produce a number greater than 1 X 10^1000, I wouldn't be surprised if it was a lot bigger than that actually.
This concludes my "presentation." If you have any questions or comments, or if you caught a mistake that I made that I missed, feel free to reply with your thoughts.
