Combinatorics: Soccer Tournament Outcomes; r-permutations and r-combinations of sets

1. Sep 19, 2011

gbean

1. The problem statement, all variables and given/known data
In a soccer tournament of 15 teams, the top three teams are awarded gold, silver, and bronze cups, and the last three teams are dropped to a lower league. We regard two outcomes of the tournament as the same if the teams that receive the gold, silver, and bronze cups, respectively, are identical and the teams which drop to a lower league are also identical. How many different possible outcomes are there for the tournament?

2. Relevant equations
n-combinations on k objects: n choose k = n!/[k!(n-k)!]
n-permutations on r objects: n!/(n-r)!

3. The attempt at a solution
So there will be 3 teams rewarded, and 3 teams bumped down a league.

The middle 6 teams placements are not identical, so they are ordered, so use permutations to figure out how many ways to order 6 teams in 6 places: [6]6 = 6!/(6-6)! = 6! ways to order 6 teams in 6 places.

I'm having trouble figuring out the number of possibilities with the first and last 3 teams.