Permutations Polynomials and PowerPoint

1
Permutations Polynomials and PowerPoint

• Leura 2001
• Note that this is about one third of the
  original. Comments to
• mneedlem_at_bigpond.net.au

2
What are we going to do?

• Find out about derangements.

• See a different PowerPoint approach.

• Find out about polynomials.

3
What do you have to do ?

• Watch out for Buffy.

4
What do you have to do ?

• Watch out for Buffy.
• When the music starts you should be writing!

5
Derangements

• A permutation is an arrangement.

• Write down all the permutations of a,b,c

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Derangements

• A permutation is an arrangement.

• Write down all the permutations of a,b,c

7
Derangements

• A permutation is an arrangement.

• Write down all the permutations of a,b,c

a, b, c,a, c, b,b, a, c,b, c, a,c, a,
b c, b, a
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Derangements

• Two of our permutations are derangements.

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Derangements

• Two of our permutations are derangements.

a, b, c,a, c, b,b, a, c,b, c, a,c, a,
b c, b, a
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Derangements

• Two of our permutations are derangements.

a, b, c,a, c, b,b, a, c,b, c, a,c, a,
b c, b, a

• Derangements are permutations in which no item
  occupies its original position.

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Derangements

• Two of our permutations are derangements.

a, b, c,a, c, b,b, a, c,b, c, a,c, a,
b c, b, a

• Derangements are permutations in which no item
  occupies its original position.

So the number of derangements of 3 items is 2.
12
Derangements

• Two of our permutations are derangements.

a, b, c,a, c, b,b, a, c,b, c, a,c, a,
b c, b, a

• Derangements are permutations in which no item
  occupies its original position.

So the number of derangements of 3 items is 2.
We write this as ?(3) 2
13
Derangements

• Two of our permutations are derangements.

a, b, c,a, c, b,b, a, c,b, c, a,c, a,
b c, b, a

• Derangements are permutations in which no item
  occupies its original position.

So the number of derangements of 3 items is 2.
We write this as ?(3) 2
14
2 elements

• Write down all the permutations of a,b

• Use your answer to write down ?(2)

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2 elements

• Write down all the permutations of a,b

• Use your answer to write down ?(2)

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1 element

• What would be the value of ?(1) ?

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1 element

• What would be the value of ?(1) ?
• ?(1)0
• Since if you start with one element it must
  occupy its original position

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1 element

• What would be the value of ?(1) ?
• ?(1)0
• Since if you start with one element it must
  occupy its original position

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4 elements

• The problem is that there are 24 permutations.

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4 elements

• The problem is that there are 24 permutations.
• a,b,c,d, a,b,d,c, a,c,b,d, a,c,d,b,
  a,d,b,c, a,d,c,b
• b,a,c,d, b,a,d,c, b,c,a,d, b,c,d,a,
  b,d,a,c, b,d,c,a
• c,a,b,d, c,a,d,b, c,b,a,d, c,b,d,a,
  c,d,a,b, c,d,b,a
• d,a,b,c, d,a,c,b, d,b,a,c, d,b,c,a,
  d,c,a,b, d,c,b,a
• Your mission (should you choose to accept it ..)

21
4 elements

• The problem is that there are 24 permutations.
• a,b,c,d, a,b,d,c, a,c,b,d, a,c,d,b,
  a,d,b,c, a,d,c,b
• b,a,c,d, b,a,d,c, b,c,a,d, b,c,d,a,
  b,d,a,c, b,d,c,a
• c,a,b,d, c,a,d,b, c,b,a,d, c,b,d,a,
  c,d,a,b, c,d,b,a
• d,a,b,c, d,a,c,b, d,b,a,c, d,b,c,a,
  d,c,a,b, d,c,b,a
• Your mission (should you choose to accept it ..)
• Is to write down only the derangements.

22
4 elements

• The problem is that there are 24 permutations.
• a,b,c,d, a,b,d,c, a,c,b,d, a,c,d,b,
  a,d,b,c, a,d,c,b
• b,a,c,d, b,a,d,c, b,c,a,d, b,c,d,a,
  b,d,a,c, b,d,c,a
• c,a,b,d, c,a,d,b, c,b,a,d, c,b,d,a,
  c,d,a,b, c,d,b,a
• d,a,b,c, d,a,c,b, d,b,a,c, d,b,c,a,
  d,c,a,b, d,c,b,a
• Your mission (should you choose to accept it ..)
• Is to write down only the derangements.

23
4 elements

• There are 9 derangements.
• a,b,c,d, a,b,d,c, a,c,b,d, a,c,d,b,
  a,d,b,c, a,d,c,b
• b,a,c,d, b,a,d,c, b,c,a,d, b,c,d,a,
  b,d,a,c, b,d,c,a
• c,a,b,d, c,a,d,b, c,b,a,d, c,b,d,a,
  c,d,a,b, c,d,b,a
• d,a,b,c, d,a,c,b, d,b,a,c, d,b,c,a,
  d,c,a,b, d,c,b,a
• So ?(4) 9

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The story so far..

• ?(1) 0
• ?(2) 1
• ?(3) 2
• ?(4) 9
• and you get one for free ..
• ?(5) 44

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The story so far..

• ?(1) 0
• ?(2) 1
• ?(3) 2
• ?(4) 9
• ?(5) 44

Find the pattern and write ?(6) to be continued.