If you are filling out a bracket for the 2010 World Cup , here are the number of ways a bracket can go:

8 groups of 4 teams each. Top 2 move on, and the order is important. So that is 4 teams choose 2 teams or 6 ways to pick the two teams. Then, those 6 pairs of teams can be ordered 2 diff. ways, so 12 possibilities per group. There are 8 groups, so 12^8 possible ways to pick the top 2 teams in order from the 8 groups. Then, these 16 teams play a single elim tourney which is 15 games, so 2^15 possible ways that can go.

This gives 12^8 * 2^15 which is 14089640214528 (14 trillion or 1.4 x 10^13) ways for the tournament to go. Which of these ways do you hope or expect it to go?

(Btw, for comparison purposes, the NCAA basketball tournament with 64 teams has 2^63 or 9223372036854775808  (9 quintillion or 9.22 x 10^18 ) possibilities. )

Now, given the way the knockout stage works, you could have symmetries with respect to which teams play which. We don’t have to worry about those if we define our problem to also take into consideration the days and stadiums in which the teams play.