# This one may be a doozy, All Possible Card Combinations of a Deck of Cards

Alright, so here goes, I'm trying to figure out all possible combinations for a $60$ card deck.

For any who wish to know, the cards in question are Magic: the Gathering. (I'll avoid the use of jargon for those who don't know what it is)

Right then. So, a standard deck of cards has $60$ cards, divided into two portions Played cards and Land cards (Land cards are used to play the played cards) There are usually anywhere from $20$ to $24$ land cards, and by extension $36$ to $40$ played cards. There can be no more than $4$ of any type of card in the deck, except for $12$, two played cards (which are omitted) and $10$ "basic" lands, these can have any number in any deck.

We'll start off with one of the more common of the bunch, $36$ to $24$.

Now, we know there are $13,707$ possible played cards and $470$ possible land cards (which aren't "basic") and $10$ possible "basic" lands.

There can be up to $4$ of any card, so for this instance, there can be anywhere from $36$ to $9$ different played cards and anywhere from $1$ to $24$ different land cards.

From here, I'm a bit confused, I believe Integrals play a part here, And I hope this is enough information to solve it.

• Can you clarify the sentence "There can be no more than 4 of any type of card in the deck, except for 12, two played cards (which are omitted) and 10 "basic" lands, these can have any number in any deck."? – Alex Kruckman Oct 11 '13 at 16:46
• Note that picking the lands is independent from picking the spells; and both have easy generating functions – user7530 Oct 11 '13 at 16:52
• Please be more clear in the description of the allowable decks. How many types of played card are there? How many types of land card? What are the limitations on the quantity of each-I can't understand the part starting with "except for 12". I doubt there are 13,707 types of played card and 470 types of land card-where do these come from. Finally, does order matter, or do you just want the number of unordered decks? – Ross Millikan Oct 11 '13 at 17:09
• @AlexKruckman Any single one of the 13,707 played cards nor the 470 land cards can either have 0, 1, 2, 3, or 4 copies of itself within one deck, but no more, a deck cannot have 5 or more copies of a single card. The two omitted cards are played cards that allow more than 4 copies of itself to exist at a time within a deck. The 10 "basic" land cards (5 snow coverd, 5 standard) can have any number of themselves within a deck, this means you can have 20 of the same land card (called Monocolored, since they are indeed colored). – vbcnxm Oct 11 '13 at 17:29
• @RossMillikan The types of played and land cards are irrelevant, since a deck can have any combination, there are types like Creatures and Sorceries, but a deck can be built completely with or without any single type. The 12 cards that are exception to the "no more than 4" rule are either "basic" lands, a staple resource, or have special wording stating otherwise. There are in fact 13,707 different cards, the game has been around for a while, each card is its own individual identity having it's own rules, costs, and effects (though some are effective copies), and no, Order does not matter. – vbcnxm Oct 11 '13 at 17:35

With so many types, I would focus on the number of partitions of $36$ (say) into parts of no more than $4$. This is the same as the number of partitions into up to four parts, of which there are $1+18+108+351=478$. For a given partition, say $4,4,3,3,3,3,3,2,2,2,2,1,1,1,1,1$ you can select the types of card in ${17303 \choose 2,5,4,5}=92752573920480357038017777741678213984004005276859139056522400\approx 10^{62}$ ways. It will be a mess