HMMT 十一月 2010 · GEN2 赛 · 第 5 题

HMMT November 2010 — GEN2 Round — Problem 5

[ 7 ] 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? Unfair Coins

解析

[ 7 ] 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? Answer: 4015440 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 ( ) 11 of eleven players. If it does not field a new player, then it has = 462 choices. If it does field a 5 ( ) 11 new player, then it has 100 choices for the new player, and = 330 choices for the 4 other players, 4 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 . Unfair Coins