Combinations with Irregular Repetition

[Strymon]

Hey folks, I believe this is the correct thread for this topic. I was debating between general mathematics and statistics. I believe it falls here.

So, I play the card game Magic: The Gathering, and one of the core tenants of the game is building a deck from a large variety of cards. The deck is then randomized (shuffled) and the actual game involves doing your best with the cards you've drawn, as well as knowing what you are likely to draw (reminiscent of poker.) Interestingly, in building a deck, knowledge of statistics is very useful as some cards you may want to see more than others.

One thing I've always been curious of, though, is how many possible decks can be created. It would seem like a simple combination problem, or even Combination with Repetition, except for a problem. Most cards you may have no more than four of in your deck, but there are five cards that you may have any number of in your deck.

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So, here's the question:

A deck consists of 60 cards.
There are 1729 cards of which you may have 0 to 4 copies each in your deck.
There are 5 cards of which you may have any number of copies in your deck.

How many unique decks can be built?

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More than the answer (though I'd be excited to see it,) I'd like to know how to frame an equation that would handle a situation like this. Any thoughts are welcome!

Thanks!

 

[mfb]

If the problem is too complex to find a direct formula, split it in sub-problems:

Assume N of those 60 cards are used which cannot appear more than 4 times, and M different cards of that type are used ($M\geq \frac{N}{4}$).
This gives well-known problems for the number of combinations for those N cards and the 60-N other cards. I am too lazy to look up the formulas.
Afterwards, you can sum over all N and M.

 

[Strymon]

Mfb, thanks for your reply!

I actually was not aware there were formulas for combination with limited repetition like this. I had searched around online briefly but wasn't able to find anything that would quite work, even as a partial solution. Are there some particular keywords I could search under to find them?

 

[bpet]

One way is using generating functions, so the solution is the coefficient of x^60 in (1+x+x^2+x^3+x^4)^1729/(1-x)^5

 

[blue_raver22]

I find this topic very interesting and relevant to the field of statistics and probability. In this case, we are dealing with combinations with irregular repetition, which is a common problem in many real-world scenarios such as card games, stock portfolios, and even genetic combinations.

To answer the question, we first need to understand the concept of combinations with repetition. In this case, we have 1729 cards, and we can choose 0 to 4 copies of each, which gives us a total of 5 options for each card. This means that the number of possible combinations for these cards is 5^1729.

However, we also have 5 cards that can have any number of copies in our deck. This means that for each of these 5 cards, we have an infinite number of options, which makes the calculation more complex.

To solve this problem, we can use the concept of generating functions, which is a powerful tool in combinatorics. A generating function is a power series that represents a sequence of numbers, in this case, the number of possible combinations for each card. By multiplying these generating functions together, we can get the total number of possible combinations for the entire deck.

Without going into too much detail, the equation for this problem would be:

(1+x+x^2+x^3+x^4)^1724 * (1+x+x^2+x^3+x^4+x^5+x^6+...)^5

Where the first part represents the 1724 cards with 0 to 4 copies each, and the second part represents the 5 cards with any number of copies. By expanding this equation and simplifying it, we can get the total number of unique decks that can be built.

In conclusion, the answer to the question would be a very large number, and it would be difficult to calculate it without using generating functions. This problem highlights the importance of understanding statistics and probability in various real-world scenarios, and how they can be applied to solve complex problems.