HMMT 十一月 2010 · GEN2 赛 · 第 5 题

英文解析

5. [ 7 ] There are 111 Star craft progamers. The Star craft team SKT starts with a given set of elevenprogamers on it, and at the end of each season, it drops a progamer and adds a progamer (possiblythe same one). At the start of the second season, SKT has to field a team of five progamers to playthe opening match. How many different lineups of five players could be fielded if the order of playerson 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
   ( )
   of eleven players. If it does not field a new player, then it has = 462 choices. If it does field a 11
   ( )5
   new player, then it has 100 choices for the new player, and = 330 choices for the 4 other players,11
   giving 33000 possibilities. Thus, SKT can field at most 33000 + 462 = 33462 unordered lineups, and 4
   multiplying this by 120, we find the answer to be 4015440 .
   Unfair Coins