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So how many possible Commander decks are there? Let's find out!
- The finite number of EDH legal cards (@scryfall f:edh lists over 19,000 cards that are legal in Commander, and if you're not using Firefox and a custom Scryfall search engine, you really ought to try it, as it's a massive quality of life improvement).
- The firm upper bound on the number of cards in a deck. There are infinitely many Standard decks because Standard has no maximum deck size. I can always add a basic land to those decks to produce a new decklist.
Let's start by getting some statistics.
|Color Identity||Commanders||Partner Pairs*||Singleton Cards||Unlimited Cards||Can Run [[Seven Dwarves]]|
I am not taking Companions into account for the following reasons:
- It makes the math much harder. I'd have to take into account the decks that could run each of the Companions and then list them with and without the partner.
- Meeting the requirements of the EDH-legal companions is difficult to do and end up with a functioning deck.
- I absolutely hate the Companion mechanic, to the extent that I refuse to purchase any card from a product that uses it. (It doesn't help that it is patently obvious that Wizards mad no effort at all to playtest Ikoria for any constructed format.)
First, a note on notation: when you see two numbers above each other, that's a standard notation for a Combination of the top number of things taken the bottom number of times without replacement. Read the Wikipedia article. Unfortunately, this subreddit does not contain TeX-style formatting. However, we're going to be taking the sum of four series:
- The series of all decks with single commanders and not running Seven Dwarves
- The series of all decks with single commanders that do run Seven Dwarves (so Red must be an element of the color identity)
- The series of all decks with two commanders that do not run Seven Dwarves
- The series of all decks with two commanders that do run Seven Dwarves.
Alternately, for those that aren't comfortable with that level of mathematical notation, here's a little bit of rectally-extracted Python (it's ugly, and I'm sure if I cared more about it I could make it better):
#!/usbin/env python3 import operator, functools #This function performs a combination of n things taken r at a time def ncr(n, r): r = min(r, n-r) numerator = functools.reduce(operator.mul, range(n, n-r, -1), 1) denominator = functools.reduce(operator.mul, range(1, r+1), 1) return numerator // denominator #This computes the number of decks with one commander and no copies of Seven Dwarves def oneCommanderNoDwarves(commanders, singletons, unlimiteds): sum = 0 for i in range(0, 100): sum += commanders * ncr(others - 1, i) * unlimiteds ** (99-i) return sum #This computes the number of decks with one commander and some number of Seven Dwarves def oneCommanderWithDwarves(commanders, singletons, unlimiteds): sum = 0 for s in range(1, 8): #range goes to one less the top number in Python for i in range(0, 100): sum += commanders * ncr(others - 1, max(i-s, 0)) * unlimiteds ** (99-s-i) return sum #This computes the number of Partner decks that do not use Seven Dwarves def partnerWithoutDwarves(partners, singletons, unlimiteds): sum = 0 for i in range(0, 99): sum += partners * ncr(others - 2, i) * unlimiteds ** (98-i) return sum #This computes the number of Partner decks that use Seven Dwarves def partnersWithDwarves(partners, singletons, unlimiteds): sum = 0 for s in range(1, 8): for i in range(0, 99): sum += partners * ncr(others - 2, max(i-s, 0)) * unlimiteds ** (98-s-i) return sum #This combines the above def allDecks(colorIdentity, commanders, partners, singletons, unlimiteds): decks = oneCommanderNoDwarves(commanders, singletons, unlimiteds) if partners > 0: decks += partnerWithoutDwarves(partners, singletons, unlimiteds) if "R" in colorIdentity: decks += oneCommanderWithDwarves(commanders, singletons, unlimiteds) if "R" in colorIdentity and partners > 0: decks += partnersWithDwarves(partners, singletons, unlimiteds) return decks #Let's just pseudocode this for right now. if __name__ == "__main__": table = #open the table above with csv.DictReader decks = 0 for row in table: decks += allDecks(row["Color Identity"], row["Commanders"], row["Partner Pairs"], row["Singleton Cards"], row["Unlimited Cards"]) print(decks)The number is big. Very big. Just as a case study, let's take the Colorless decks. Doing the math on just them leads to a number around 5.254 * 10166. For comparison, the age of the universe in seconds is somewhere around 4.35 * 1017 seconds.
So are you ready for this? The number of possible decklists excluding companions is 8.583 * 10 270 as of 2020-07-10. New sets will definitely increase this number considerably. The problem of enumerating these decks is intractable, but the process will eventually halt.
Now here's one other thing I should note: the vast majority of these decks are unplayable or at least functionally unplayable, despite being legal lists. Why? Because any deck, without respect to color identity, has a lot of cases where it isn't running enough mana sources to make its colors reliably. In fact, it may wind up that the only mana sources in the deck are basic Wastes (which can go in any number in any deck), if it has that many. But this question wasn't about playability. It was about how many possible valid lists there are.
* I am including the pairing of [[Gisela, the Broken Blade]] and [[Bruna, the Fading Light]], as allowing these two to work as though they had Partner With is sensible, flavorful, intuitive, a common house rule, and why the hell doesn't Commander have the rule, "If two legendary creatures can meld together, they may both be your commander as though they had Partner with each other." It's the single rule change I'd make to the format.