Lesson 22: Pascal

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Lesson 22 Pascals Triangle and Permutations

• Objectives
• State Pascals Identity
• Diagram Pascals Triangle
• State Vandermondes Identity
• Determine Permuations with repetition
• Determine Permutation with indistinguishable
  objects

• Outline
• Pascals Identity and Triangle
• Vandermondes Identity
• Permutations (general)
• Indistinguishable objects
• HW Due 4/21
• 4.3 3, 5d-f, 6d-f, 10, 18, 21ace, 24
• 4.4 4, 9
• 4.5 5, 10, 16, 22, 29, 35, 44

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Pascals Identity

• If n and k are positive integers w/ nk

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Pascals Triangle
1
1 1
1 2 1
http//www.cs.washington.edu/homes/jbaer/classes/b
laise/blaise.html
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Vandermondes Identity

• Let m, n, r be nonnegative integers with r less
  than or equal to m and n.

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Problem 1

• The 6th row of Pascals Triangle is
• 1 5 10 10 5 1
• what is the next row?

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Problem 2

• Write a recursive algorithm to calculate n choose
  k
• int choose ( int n, int k)

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r-permutations

• If repetition is allowed, a set of n objects has
  nr permutations
• Ex How many strings can be formed with the Greek
  uppercase letters of size n?
• From 1-n?

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In Class Riddle
How many ways can you select five bills from a
cash box having 1s 2s, 5s, 10s, 20s,
50s, and 100s?
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Combinations with Repetition

• There are C(nr-1, r) r-combinations from a set
  with n elements when repetion of elements is
  allowed
• How many ways can you select five bills from a
  cash box having 1s 2s, 5s, 10s, 20s,
  50s, and 100s?

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Example 1

• A bakery sells four kinds of cookies chocolate,
  jelly, sugar, and peanut butter. You want to buy
  a bag of 30 cookies. Assuming that the bakery has
  at least 30 of each kind of cookie, how many bags
  of 30 cookies could you buy if you must choose
• (a) at least 3 chocolate cookies and at least 6
  peanut butter cookies.
• (b) exactly 3 chocolate cookies and exactly 6
  peanut butter cookies.
• (c) at most 5 sugar cookies.
• (d) at least one of each of the four types of
  cookies

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Indistinguishable Permutaions

• The number of different permutations of n objects
  where there are n1 indistinguishable objects of
  type 1, n2 indistinguishable objects of type
  2.... and nk indistinguishable objects of type k
  is
• n!/n1!n2!...nk!

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Example 2

• In how many ways can the letters in DECEIVED be
  arranged in a row?

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Indistinguishable Permutaions

• The number of ways to distribute n
  distinguishable objects into k distinguishable
  boxes so that box i contains ni elements
• n!/n1!n2!...nk!

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Distinguishable objects into boxes

• 20 people show up for a tournament. How many
  ways are there to have 4 teams of 5?
• Three teams of 6?

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Summary
r-permutations no
r-combinations no
r-permutations yes
r-combinations yes
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Summary Examples

1. There are 24 people in a jury pool. How many
   possible juries can be selected?
2. There are 15 people in a group. How many ways
   are there to order 4 in a line for a picture?
3. How many strings are there of length 4?
4. There are 6 kinds of fruit to choose from. How
   many fruit baskets of a dozen fruit are there?

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Problem 3

• How many ways are there to choose a dozen donuts
  in a bakery that has 21 kinds?

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Problem 4

• A bagel shop has 8 kinds of bagels. How many
  ways are there to choose a dozen bagels if you
  must have at least one of each kind?

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Problem 5

• How many ways are there to make a string from the
  letters in MISSISSIPPI, if you must use all of
  the letters?