How many different hero combinations in 5v5 game are there?

[nitko]

Hi Dota players. At the moment there are 112 heroes in the map. Anyone ever wondered how many different hero setups exists? Disregard for a moment that some heroes are vastly more popular then others, lets say mode is AR. I just calculated it and its damn huge number. Will not tell you yet, lets see first some guesses (or an answer) from dota vets or forum regulars. :D

[jeandarc]

Too lazy to do the math atm xd

Its not really hard, its just long but easy

[Cen]

If it matters which player owns the hero (aka, if hero is in different slot it's a different variation):
n!/(n-k)! = 112!/(112-10)! = 205368317946676684800

If it does not matter and we only care which heroes were picked:
n!/k!*(n-k)! = 112!/10!*(112-10)! = 56594002961496

If you ask cen:
1 because Spec ftw!

[nitko]

hmmm.

No it doesn't matter which exact slot got which hero. Team with "A" "B" "C" "D" "E" heroes is == as team playing "C" "D" "E" "B" "A" heroes.

Your second calculation is strange. It doesn't even compute to what you wrote (5.6 x 10^13) but to some vastly bigger number with 337 digits.

Maybe it doesn't say much but my calculation is somewhere in between of those 2 numbers :D

[Cen]

http://www.wolframalpha.com/input/?i=112!%2F%2810!*%28112-10%29!%29

[nitko]

Yep, that looks right. Basically we are picking 10 from the set of 112. Without factorials, can be simply written as: (112*111*110*109*108*107*106*105*104*103) / (1*2*3*4*5*6*7*8*9*10) which results to:

>>> 56 594 002 961 496 <<< as Cen actually wrote in first post.

In English that reads as 56 trillion 594 billion 2 million 961 thousand 496 differnet hero combinations.

For comparison, there is "only" something like 15 million of different combinations in standard lottery (7/39) so that makes around million times more chance that you will guess/win lottery then which 10 heroes will be randomed in dota AR match.

And every AR game in the history of dota is guaranteed to have/been its own unique hero setup (implying that game/map can actually generate random numbers every time and there are no pre-seeding; I'm personally not sure how randomization in dota map exactly is done, tech wise).