(If I didn't make mistakes - which I'm sure I did)
Mistakes on two fronts. You want to include doubles for number of teams, and exclude doubles for number of matchups (doubles in matchups are 50-50). Also, once you have your total number of teams, and you want to consider matchups, raising to a power of two and subtracting the number of teams wasn't the right way to go. Here's a quick hint. For any one matchup considered, do you not
choose two teams per matchup?
When choosing 2 from a set, you don't need factorials. You can solve choose 2 problems with a series. I'll give the standard formulas at the bottom, but for now I'll solve using series', as that makes it clear what the difference between doubles and no-doubles is, and why we need to use both to figure total matchups.
First, let's show that using a series will work for both inclusive and exclusive choose 2 problems using word examples. How many sets of two letters can you make from the word CAT including doubles, and excluding doubles?:
CAT choose 2 inclusive (with doubles):
CC, CA, CT, AA, AT, TT = 6 combos, 3 start with C, 2 start with A, one starts with T. (3+2+1)
CAT choose 2 exclusive (without doubles):
CA, CT, AT = 3 combos, 2 start with C, one starts with A. (2+1)
You get the
exclusive result when you run 3 choose 2 on a calculator.
The difference between inclusvie and exclusive = 3 which is the same number of letters in CAT (X=3)
Now let's try the same thing with the word "STOP"
STOP choose 2 inclusive:
SS, ST, SO, SP, TT, TO, TP, OO, OP, PP = 10 combos (4+3+2+1)
STOP choose 2 exclusive:
ST, SO, SP, TO, TP, OP = 6 combos, (3+2+1)
You get the exclusive result when you run 4 choose 2 on a calculator.
Difference between inclusive and exclusive = 4 which is the same number of letters in STOP (X=4)
This pattern holds watter no matter how high you go, so long as you're only doing choose 2. So, now that we know we can, let's find the total number of matchups in SSBB doubles using series'. I'll write out the formulas in an exemplarity notation (the same notation I saw it in when my instructor was introducing us to these kinds of problems):
35 choose 2
inclusive is the number of possible character combinations (including doubles).
Calculators will typically run the exclusive formula by default (see word examples for proof). Because we're choosing 2, fixing this is easy if you want to run it on your calculator. Simply add X to your result after your calculator gives you your answer. If that doesn't make sense, look up at the word examples. (see above word examples for proof)
Here's the inclusive formula:
(X+X-1+X-2+X-3.....+1) where X is the number of objects chosen from.
This formula INCLUDES double picks. Remember, the difference between exclusive and inclusive (no doubles and doubles) is X. Since calculators give us exclusive, all we need to do is add X for choose 2 inclusive problems.
35+34+33+32........+1 = 630
OR
Calculator: 35choose2 = 595 +35 = 630
Now that we know how many team combos there are, we want to find the number of matchups. For any one game, we're
choosing 2 teams, which means that we're going to run a choose formula again, only this time we DON'T want to be able to pick two of the same object (team) on any one pick (same team vs same team is 50-50). So now we use the standard
exclusive formula (the only difference between the two formulas is that we start on X-1 instead of starting on X).
Here's the exclusive formula:
(X-1+X-2+X-3.....+1) where X is the number of objects chosen from. In our case X= 630.
629+628+627+626+.........+1 = 198,135 total team matchups.
or just run it on your calculator,
this time without adding X.
Calculator: 630choose2 =
198,135
And that's the grand total number of matchups that would need to be considered in a SSBB Doubles tier list (
198,135)
One-hundred ninety-eight thousand, one-hundred thirty-five total matchups to consider, which is why there's no doubles tier list.
Interesting tidbit of info, is that for 1v1 matchups, the number of matchups would be 35 choose 35 exclusive, which is 595.
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I was using series formulas to solve choose 2 problems to explain things. You can do without the standard formulas for choose 2, but for any more you need a standard choose formula (or a permutations formula with p! included in the denominator).
Standard exclusive formula (the one the calculator will give you).
n choose p = n! / p!(n-p!)
Inclusive formula.
n choose p = (n! / p!(n-p!)) + (n(p-1))
You can still get an inclusive result easily from a calculator. Just have to add n(p-1).
So, for total teams (running inclusive formula):
35 choose 2
=(35)! / (2)!((35)-(2))! + ((35)((2)-1))
=35! / 2*1(33)! + 35
=(35*34) / 2 + 35
=630
And now for number of matchups (running exclusive formula):
630 choose 2
=(630)! / (2)! ((630)-(2))!
=630! / 2(628)!
=630*629 / 2
=198,135
