Let A be a set of 16 football teams and N be a set of 16 different football teams. So A ∩ N is empty. Let set L = A U N. 1. Seven teams from A make the playoffs. How many different sets of 7 teams from A could make the playoffs? 2. At the beginning of the playoffs, there are 7 teams from A and 7 teams from N remaining. The Super Bowl pairs a team from A with a team from N. How many different Super Bowl pairings are there at the beginning of the playoffs? Briefly justify your answer. 3. The result of the Super Bowl can be represented by an ordered pair of the teams playing, where the first team is the winner and the second team is the loser. For example, if the Dolphins defeat the Panthers in the Super Bowl, the result is (Dolphins, Panthers). How many different possible results of the Super Bowl are there at the beginning of the playoffs? Briefly justify your answer.
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
Let A be a set of 16 football teams and N be a set of 16 different football teams. So A ∩ N is empty. Let set L = A U N.
1. Seven teams from A make the playoffs. How many different sets of 7 teams from A could make the playoffs?
2. At the beginning of the playoffs, there are 7 teams from A and 7 teams from N remaining. The Super Bowl pairs a team from A with a team from N. How many different Super Bowl pairings are there at the beginning of the playoffs? Briefly justify your answer.
3. The result of the Super Bowl can be represented by an ordered pair of the teams playing, where the first team is the winner and the second team is the loser. For example, if the Dolphins defeat the Panthers in the Super Bowl, the result is (Dolphins, Panthers). How many different possible results of the Super Bowl are there at the beginning of the playoffs? Briefly justify your answer.
4. At the beginning of the playoffs, how many results of the Super Bowl are there where the Dolphins (from set A) do not win the Super Bowl? You can assume that the Dolphins are among the 7 playoff teams from their set A. Briefly justify your answer.
5. During the season the Dolphins will play five consecutive games against other teams from A. Exactly two of the games will be against the Patriots (who are in A), while the remaining three games are against to-be-determined unique teams. There must be exactly two games in between the games against the Patriots. How many different possible schedules are there? Note that the Dolphins cannot play against themselves. Justify your answer.
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