How Many Team Comp Possibilities? ANSWERED! (Strategic Diversity - Please Check My Math)

UPDATE: Per my comment in response to ZeusHasRefresher, if you limit yourself to meta picks only, the number of 5v5 combinations drops to 32,035,809,028,608 (32 trillion).

OK. So I think one of the reasons we all love LOL so much is because of the incredible variety of compositions that make most games feel unique. (Granted, you see a lot of the same supports and adcs, etc. But still, there's a lot of variety).

I wanted to calculate the theoretical variety in a 5v5 ranked match (meaning no champ used twice, unlike blind pick).

To start, we can gather how many possible combinations of 10 there are with the current 127 champs. This is pretty straightforward. For this discussion bans don't matter, because for any lobby where a particular 6 champions are banned, we can imagine other lobbies where they are not, such that any combination of 10 champs is possible.

So there are 127 possible first picks. With one champ off the table, there are 126 possible 2nd picks. That means in the first 2 picks alone there are 127 * 126 possible pick outcomes, or 16002 possibilities.

So to draft 10 champs there are 127 x 126 x 125 x 125 x 123 x 122 x 121 x 120 x 119 x 118 possible draft orders. That's 764,988,346,351,845,000,000 possible drafts, or for simplicity, just shy of 765 quintillion! If you consider every role on every side unique (e.g. having a red side jungle is different than having a blue side jungle which is considered different than having a blue side top , etc.), this is how many games could be played.

If every game lasted exactly 30 minutes, and you never slept, it would take you over 43 quadrillion years to play a game with every single one of these compositions. That's about 3 million times the age of our universe. It's a long time.

If you find this post in the future and the game has changed (more champs overall, more or fewer champs per game, etc.), or if you want to figure out the options for a different mode, here's the formula.

p = c!/(c-s)!

Where: c = number of champions s = number of summoners in each game p= possibilities ! = the factorial operation, which is a number *(number -1) *(number - 2).... all the way to 1.

If you've got Excel handy, you can just paste this into a cell, replacing c and s with current values:

=fact(c)/fact(c-s)

However, suppose you just want to know how many unique 5v5s there are, not all the permutations they could take. For this problem, if the team comp is , , , , and , we don't care if they are on blue side or on red side, and we don't care if goes jungle, top or otherwise. Any combination of those 5 champs is considered a single possibility.

Well that's straightforward enough, first we just just need to subtract out all the repetitive combinations. In a group of 10, the repetitive combinations are 10! (That's 10 factorial, not 10 I'm so excited). So we modify our formula to:

p = c!/(c-s)!/s!

For Excel:

=fact(c)/fact(c-s)/fact(s)

At present writing, this gives the result of 209,123,798,385,425, almost 210 trillion.

We're not quite done yet. 210 trillion is the number of unique combinations of 10 without repetition. In that setup, if blue is:

and red is:

we would not consider it a new combination if blue was:

,

and red was:

While this may satisfy some, it doesn't satisfy me. So now that we know how many unique combinations of 10 there are, we need to find out how many unique combinations of 5 man teams there are for each group of 10.

This turns out to look exactly the same as the formula we've already used just with much smaller numbers:

k = s!/((s-t)! * t!)

Where: k = unique compositions (avoided c to eliminate confusion with prior formulas) s = number of summoners in each game (synonymous with prior use of s) t= number of members on each team

For games with 10 people forming teams of 5, that gives us 252 possibilities for each 10 man comp.

Now all we have to do is multiply the number of unique 10 man games by the number of unique 5 man comps in each of those games, and we get 52,699,197,193,127,200. That's well over 50 quadrillion combinations.

Here's a master excel formula to get you to that same number in the future. I've already simplified by reducing common terms:

=fact(c)/fact(c-s)/fact(s-t)/fact(t)

Sure, you're probably never gonna play a game with

vs.

.

However, when you factor in summoner spells, items, and skill order, the possibilities get even more ridiculously large.

Just for giggles, what happens when league releases the 128th champion? It will increase the number of unique 5v5 comps by about 4.5 quadrillion.

Punchline: as long as Riot continues to create new champions and buff/nerf existing champions in the pursuit strategic diversity, it'll be a long time before any of us get bored, like 43 quadrillion years.

See you on the rift!

Side note: If you want a review of the math concepts that went into this, check out https://www.mathsisfun.com/combinatorics/combinations-permutations.html