Permutations and Combinations

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Permutations and Combinations

• Making your own starting lineup!

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Why do we want to study permutation and
combination?

• Very often we will find situations where one
  thing can be done in different ways.
• In order to find the best way, we need to know
  how many possible ways are there in total.
• For example, computer A send message to computer
  B through the network. How many possible routes
  are there? Which one is the best?

A
B
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Activity 1 Scissor Paper Rock

• Two students play scissor paper rock
• How many possible ways the play will go?

Student 1
Scissor
Paper
Rock
Student 2
Scissor
Paper
Rock
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Activity 2 Group play

• Two groups of students. Each time one student
  from each group play scissor paper rock against
  each other until all students have played.

If you have 4 students in the group, how many
different lineups can you possibly make?
Student 1
Student 2
Student 3
Student 4
Student 1
Student 3
Student 2
Student 4
Student 4
Student 3
Student 2
Student 1
, , , ,
Permutation. 4!24
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What if you only have 3 students in the group,
but you are playing against a group with 4
students? How many different lineups can you
possibly make?
each of the 3 students have to play in each of
the first 3 rounds, the 4th round you can assign
any of the 3 to play again.
Student 1
Student 2
Student 3
Student 1
Student 1
Student 2
Student 3
Student 2
Student 3
Student 2
Student 1
Student 3
, , , ,
First 3 spots Permutation. 3!6Last spot
3.Total 6x318.
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Activity 3 Combinations

• A group of n students (ngt3), pick 2 students
  from them to form a team to play against another
  team from another group.

How many possible ways you can form your team?
(Assuming the playing order within a does not
matter.)

• Combination 43/216

How many possible ways you can form your team?
(Assuming the playing order matters.)

• This goes back to permutation again 4312

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Activity 4 Combinations and Permutations

• 4 piles of cards. 2 cards in each pile
• Pick one from each pile, how many combinations of
  cards can you possibly get?
• Lets count!
• 4 piles of cards. 4 cards in each pile
• Pick 2 cards from each pile, how many
  combinations of cards can you possibly get?
• Lets calculate!

Each pile has combinations C(4,2)6 6x6x6x61296
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• 4 piles of cards. 2 cards in each pile
• Pick one from each pile in sequence, how many
  permutations of cards can you possibly get?
• Lets count!

• ?A, ?A, ?A, ?A.
• ?A, ?A, ?A, ?2
• ?A, ?A, ?2, ?A.
• 164!384

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• 4 piles of cards. 4 cards in each pile
• Pick 2 cards from each pile in sequence, how many
  permutations of cards can you possibly get?
• Lets calculate!

1. Assume the choice of pile sequence is
   determined.Permutation within each pile P(4,
   2). Permutation of the whole sequence P(4,
   2)4
2. Now consider the permutation of 4 piles 4!
3. So total permutation 4! x P(4, 2)4.