How many combinations exist in a 64 team single elimination tournament.?
investments 101
2006-03-12 11:18:32 UTC
How many sheets of paper would I have to fill out to pick the Final 4 exactly the way it will happen? For instance if there are 2 teams there would be 1 possibility, 4 teams there are 8 possibilities. How many possibilities are there with 64 teams.
Seven answers:
2006-03-12 21:03:01 UTC
There are a total of 63 games to be played (every team will lose exactly once, except the entire winner of the tournament).
In each game, there are exactly two possible outcomes (given the results of the games leading up to it).
Therefore the number of completely-filled brackets is 2^63, that is 2 to the 63rd power, that is 2*2*2*...*2 (63 times).
This is an *enormous* number. To give you an idea, it is 19 decimal digits long.
In general, if there are n teams, then there are 2^(n-1) ways to fill out the bracket completely. This formula matches your examply (although, with 2 teams there are actually 2 possibilities).
2016-03-16 05:56:21 UTC
If you reduce this problem to a one round match between beginner vs expert. The answer is Y = p. If you start with 4 players - 2 begs, 2 exps, then the answer is Y = p/3 * [1 + 2p*(3-2p)] This comes from a simple probability tree. Note that in both cases, Y {| p = 0} = 0 Y {| p = 1/2} = 1/2 Y {| p = 1} = 1 Also Y(p) + Y(1-p) = 1 This is supposed to happen. In fact you expect odd symettry about Y = 1/2. You can rewrite it as Y - 1/2 = 4/3 * (p - 1/2) - 4/3 * (p - 1/2)^3 Letting u = Y - 1/2, and x = p - 1/2 u(x) = 4/3 * (x - x^3) with -u(-x) = u(x) and u(1/2) = 1/2 Extend it to n rounds, and you expect an answer of the form Y = p * P(p^2n-2) where P(p^2n-2) is a polynomial of order 2n-2 satisfying the conditions above. Clearly a closed form expression for Y exists, but is so tedious to find analytically, not too many computer software can handle this for variable p. However I'm willing to postulate that the answer should be of the form: u = a*x*(1 - x^2 + x^4 - x^6 + x^8 - x^10) u(1/2) = 1/2 implies that a = 1024 / 819 Postulated Answer: Y = 1/2 + 1024 /819 *∑{n=0..5} (-1)^n * (p - 1/2)^(2n+1)