June 18, 2003

1
Lecture 9

• June 18, 2003

2
Combinatorics

• The branch of mathematics studying the
  enumeration, combination, and permutation of sets
  of elements and the mathematical relations which
  characterize these properties.
• Examples of combinatorial problems
• Scheduling classes (or tournament play)
• Determining cheapest route for a multi-city tour
  
• Tracing a figure without picking up your pencil
  or repeating a line
• Determining the odds of a full house in poker
• Designing reliable networks
• Creating efficient computer programs.

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Examples

• Two dice are rolled, one black and one white,
  how many different rolls are possible? (white 1
  is not black 1)
• How many ways are there to choose three officers
  from a club of 25 people?

4
Multiplication Principle

• If a choice consists of two steps, of which
  one can be made in m ways and the other in n
  ways, then the whole choice can be made in mn
  ways.
• Example--- For any finite set S, let S denote
  the number of elements in S. If A and B are
  finite sets, then AXBAB

5
Multiplication Principle

• So we can generalize the multiplication
  principle for the multiplication of choices so
  that it applies to choices involving more than
  two steps.
• If in a sequence of n decisions, the number of
  choices for decision i does not depend on how the
  previous i-1 decisions are made, then the total
  number of ways to make the whole sequence of
  decisions is the product of the number of choices
  for each decision.
• So for k steps, where k is a positive integer, we
  have
• If a choice consists of k steps, of which the
  first can be made in n1 ways, the second in n2
  ways, , and the kth in nk ways, then the whole
  choice can be made in n1n2nk ways.

6
Addition Principle

• If a set can be partitioned into disjoint
  subsets, then the number of objects in the set  
  the sum of the number of objects in each of its
  parts.
• Example
• The number of people in this room equals the
  number of men plus the number of women.

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Addition Principle(cont.)

• The number of subsets of 1,2,3,4 equals
• the number of 0-element subsets  1
• the number of 1-element subsets 1,2, 3,4
  4
• the number of 2-element subsets 1,2, 1,3,
  1,4,
• 
  2,3, 2,4, 3,4 6
• the number of 3-element subsets 1,2,3, 1,2,4,
  
• 1,3,4, 2,3,4 4
• the number of 4-element subsets 1,2,3,4
  1
• TOTAL 16

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Addition Principle(cont.)

• A child is allowed to choose one jellybean out of
  two jellybeans, one red and one black, and a
  gummy bear out of three gummy bears, yellow,
  green, and white. How many different sets of
  candy can the child have?

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Using the principles together

• Now we can use both principles at the same
• time
• If a woman has seven blouses, five skirts and
  nine dresses, how many different outfits does she
  have?
• (75)9
• How many three-digit integers are even?

10
Decision Trees

• If the multiplication principle doesnt apply we
  can use less regular decision trees.
• We can use the multiplication principle, when the
  number of outcomes at any level of the tree is
  the same throughout that level.
• Number of outcomes of an event based on a series
  of possible choices.
• Example
• Draw the decision tree for the number of
  strings Xs, Ys and Zs with length 3 that do
  not have a Z following a Y.

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Principles in Set

• Now lets see some properties
• For any finite set S, let S denote the number
  of elements in S. If A and B are finite sets,
  then
• AXB AB
• Let A and B be disjoint finite sets.
• Then AUB AB
• If A and B are finite sets, then
• A-B A - A ? B
• and
• A-B A - B if B ? A

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Principle of Inclusion and Exclusion

• If A and B are any subset of a universal set S,
  then A-B, B-A, A ? B are mutually disjoint
  sets.
• Venn diagram representation
• What is (A-B) ? (B-A) ? (A ? B)?
• And (A-B) ? (B-A) ? (A ? B) ?
• So finally A ? B A B - A ? B

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Principle of Inclusion and Exclusion (cont.)

• So if were dealing with two sets
• A ? B A B - A ? B
• And if we are dealing with three sets
• A ? B ? C A B C - A ? B - A ? C
  -
• B ? C A ? B ? C
• What is the venn gram representation of it?

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Principle of Inclusion and Exclusion (cont.)

• So the pattern is If we have n sets, we
  should add the number of elements in the single
  sets, subtract the number of elements in the
  intersection of two sets, add the number of
  elements in the intersection of three sets,
  subtract the number of elements in the
  intersection of four sets, and so on.
• Principle of Inclusion and Exclusion

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Principle of Inclusion and Exclusion (cont.)

• How to prove it? Mathmatical induction.
• Base case
• Assumption ..
• Show . (Page 205-206)

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Principle of Inclusion and Exclusion (cont.)

• Example A survey of 150 college students reveals
  that 83 own automobiles, 97 own bikes, 28 own
  motorcycles, 53 own a car and a bike, 14 own a
  car and a motorcycles, 7 own a bike and a
  motorcycle, and 2 own all three.
• Question
• How many students own only a bike?
• How many students do not own any of the three?

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Pigeonhole Principle

• If more than k items are placed into k bins, then
  at least one bin contains more than one item.
• Example How many times must a single die be
  rolled in order to guarantee getting the same
  value twice?

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Excercise

• Exercise 3.2 28, 48, 49,58-64,69, 72
• Exercise 3.3 7, 11, 15, 19