# Combination + Permutation Question

## Homework Statement

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
kicks?

(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?

## Answers and Replies

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A) Looks right, although I doubt the goalie would be taking penalties.

B) I agree, the wording is quite ambiguous here. Its actually too confusing to even speculate. The question doesn't even tell you if you are the winning team or the losing team. nor does it tell you how many kicks have elapsed to get to that score. It could be 2-0 after 9 kicks wherein the 10th kick wouldn't even happen since the game would be finished. I've read it at least 10 times...Sorry, can't help.

I assume for B) that they are trying to keep it simple. There are 5 players taking shots on the net. If 2 players score then no more kicks are taken. I think i did this wrong. We don't want factorials, we want combinations. There is a possibility of 5C2 + 5C3 + 5C4 + 5C5 combinations of players taking shots, i think. The resulting number seems too low though. Since order is relevant, am i using the right method to calculate the total combinations? 5C2 should be more than 10 if order matters.

well if order matters it would be 5P2 , 5P3... etc. but you are neglecting the fact that the other team can score? The question states "2 MORE goals" specifically.

I am also not sure that using 5 is correct, since there are 11 possible kickers.

For your first statement of 5C2 or 5P2 (or 11C2, 11P2, whichever you decide), you would have to multiply that number by the number of ways that the other team can score 0 goals. You would also have to multiply that by the number of ways that you can score two goals.

for the next statement, you would have to multiply 11P3 by the number of ways that the other team can score 1 goal before you score 2. (since if the score is 2-0 the game is over)

This question gets complicated very very quickly, which is why I think you need a little more direction in how to tackle this.

The instructions given are pretty vague, i will admit. I doubt the teacher wants us to get too involved with the semantics of the question. Since the other team has not been mentioned I will simply assume that i am not supposed to factor in the other team's score at all.

That said, I think the way to go about calculating part B would be to firstly calculate the total number of linear combinations of 2 or more players out of the group of 5. Was i corrrect in using 5C2, etc. or am i supposed to use 5P2?

My textbook uses weird notation for combinations. It uses brackets () with 2 numbers stacked inside of them, rather than simply saying nCr. And we haven't even learned about nPr yet, to my knowledge.