Consider a team of 11 soccer players, all of whom are equally good players
and can play any position.
(a) Suppose that the team has just finished regulation time for a play-off game and
the score is tied with the other team. The coach has to select five players for
penalty kicks to decide which team wins the game. For the penalty kicks, all
five players selected for both teams take one kick each on the oppositions net;
the team with the most goals wins (assume there is a winner). Since each player
takes penalty kicks differently, the order in which the players are arranged for the
penalty kicks is important and can affect the outcome. How many different ways
(linear arrangements) can the coach select five (5) players to take the penalty
(b) Suppose that the penalty kicks are stopped after one team has two (2) more goals
than the other team. How many different ways (linear arrangements) can penalty
kicks be taken by players on a team?
(c) Assuming that all the linear arrangements in part (b) are equally likely, what is
the probability that all five players will end up taking penalty kicks?
The Attempt at a Solution
A) We have 11 soccer players and must pick a combination of 5. C(11,5) results in (11*10*9*8*7) / (5*4*3*2*1), or 462 possible combinations of players chosen to take penalty kicks. There are 5! ways to arrange each selection of players, resulting in 55440 total combinations of 5 players taking penalty kicks.
B) I am confused by the wording of this one. Are we to assume that 5 players have been chosen to take the kicks, or that there can be any 5 players taking the penalty kicks? I assume 5 players have been picked to take the kicks. Since there must be a minimum of 2 goals, there is a possible 2! * 3! * 4! * 5! combinations of players taking kicks on the net, i think.
C) To calculate this would we just subtract 4! * 3! * 2! from our previous calculation?