HMMT-Nov 2010 thm, Problem 5

There are 111 StarCraft progamers. The StarCraft team SKT starts with a given set of eleven progamers on it, and at the end of each season, it drops a progamer and adds a progamer (possibly the same one). At the start of the second season, SKT has to field a team of five progamers to play the opening match. How many different lineups of five players could be fielded if the order of players on the lineup matters?

Solution: We disregard the order of the players, multiplying our answer by $5 !=120$ at the end to account for it. Clearly, SKT will be able to field at most 1 player not in the original set of eleven players. If it does not field a new player, then it has $\left(\begin{array}{c}11 \\ 5\end{array}\right)=462$ choices. If it does field a new player, then it has 100 choices for the new player, and $\left(\begin{array}{c}11 \\ 4\end{array}\right)=330$ choices for the 4 other players, giving 33000 possibilities. Thus, SKT can field at most $33000+462=33462$ unordered lineups, and multiplying this by 120 , we find the answer to be 4015440 . \section*{Unfair Coins} $\fbox{4015440}$.

Answer: 4015440