Carc Central Community > Quizzes, Puzzles and Challenges

How many ways can you combine the official expansions

**tothederby**:

How many ways can you combine the mini and major expansions with the base game? For instance Watchtowers and the Crop Circles can be combined to form the following games: base game on its own, base game + crop circles, base game plus watchtower and base game plus watchtower plus crop circles. Answer is 4.

So if there are 38 expansions how many games can you construct in this manner?

[I have included each of the following as an expansion: Inns & Cathedrals, Traders and Builders, The Princess and the Dragon, The Tower, Abbey and Mayor, The Catapult, Bridges, Castles & Bazaars, Hills & Sheep, Under the Big Top, The Wheel of Fortune, The Fliers, The Messengers, The Ferries, The Gold Mines, The Mage & Witch, The Robbers, The Crop Circles, The Cathars / Siege / Besiegers (counted as one expansion), The Mini-Expansion GQ11, The Tunnel, The Plague, The Halflings, The School, The Phantom, The Festival, The Little Buildings, The Wind Roses, The Special Monasteries (German, Dutch, Japanese - counted as one expansion), The German Castles, The Watchtowers, The German Cathedrals, The Labyrinths, The Markets of Leipzig, The Fruit-Bearing Trees, The Barber-Surgeons and from Expansion 6 each of (counted as separate expansions): The King, The Count / City of Carcassonne and The Cult / Shrines (counted as one expansion).

I appreciate this labelling of expansions has subjectivity but the challenge is not about the labelling, but about the number of combinations that arises.]

Linkback: https://www.carcassonnecentral.com/community/index.php?topic=4238.0

**Sinscerly**:

Well there are to many ;D in worst case you use all 38 expansions and in the minimal case you use 1

Note, not counting expansions of the same expansion. :o So not twice Crop Circles in ONE game.

So 38, for one.

So 38 * 37 for two

So 38 * 37 * 36 for three

ext...

So 38 * 37 * 36 * 35 * 34 .... * 2 * 1

Now count all up and you will come to the answer, I know there is a formula for this, but I cann't remember

Dunno if correct: 523022617466601111760007224100074291200000000

Correct if I wrote it good: 1421722876932548239555534242726381983188762660

Correct one: 523022617466601111760007224100074291200000000

I was to lazy to type all this in my calculator so a simple script that runs through all

**Whaleyland**:

Sinscerly, I think your first answer was correct. The formula is ! (factorial) and I got (for 38 expansions) 52,300,226,174,666,012^28. In other words, the combinations are near-infinite.

The list below is missing The River I, The River II, Darmstadt, and five separate Spiel tiles. And Halflings I and Corn Circles I are definitely both different expansions, albeit ones with the same rules, so those get added too. Total expansions are thus 46, giving us a grand total of 5,502,622,159,812,089^42 combinations. Just a bit more. ;D

**Sinscerly**:

--- Quote from: whaleyland on March 14, 2019, 05:00:10 AM ---Sinscerly, I think your first answer was correct. The formula is ! (factorial) and I got (for 38 expansions) 52,300,226,174,666,012^28. In other words, the combinations are near-infinite.

The list below is missing The River I, The River II, Darmstadt, and five separate Spiel tiles. And Halflings I and Corn Circles I are definitely both different expansions, albeit ones with the same rules, so those get added too. Total expansions are thus 46, giving us a grand total of 5,502,622,159,812,089^42 combinations. Just a bit more. ;D

--- End quote ---

Hmm I am not sure... as it's the sum of all the combinations with 38 expansions. I thought the first one is only when I don't add them up, but I have to revisit maybe my code. ::) It's quite a difference if it's the first one :))

But I think you are right. As I see an error in my code

**tothederby**:

Thanks for the responses Sinscerly and Whaleyland.

I think its a good sized number but not as big as you are suggesting.

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