Lecture 9: Counting Principles

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Lecture 9 Counting Principles

• Discrete Mathematical Structures
• Theory and Applications

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Learning Objectives

• Learn the basic counting principlesmultiplication
  and addition
• Explore the pigeonhole principle
• Learn about permutations
• Learn about combinations

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Learning Objectives

• Learn about binomial coefficients and explore the
  algorithm to compute them
• Discover the algorithms to generate permutations
  and combinations
• Become familiar with discrete probability

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Basic Counting Principles
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Basic Counting Principles
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Basic Counting Principles

• There are three boxes containing books. The first
  box contains 15 mathematics books by different
  authors, the second box contains 12 chemistry
  books by different authors, and the third box
  contains 10 computer science books by different
  authors.
• A student wants to take a book from one of the
  three boxes. In how many ways can the student do
  this?

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Basic Counting Principles

• Suppose tasks T1, T2, and T3 are as follows
• T1 Choose a mathematics book.
• T2 Choose a chemistry book.
• T3 Choose a computer science book.
• Then tasks T1, T2, and T3 can be done in 15, 12,
  and 10 ways, respectively.
• All of these tasks are independent of each other.
  Hence, the number of ways to do one of these
  tasks is 15 12 10 37.

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Basic Counting Principles
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Basic Counting Principles

• Morgan is a lead actor in a new movie. She needs
  to shoot a scene in the morning in studio A and
  an afternoon scene in studio C. She looks at the
  map and finds that there is no direct route from
  studio A to studio C. Studio B is located between
  studios A and C. Morgans friends Brad and
  Jennifer are shooting a movie in studio B. There
  are three roads, say A1, A2, and A3, from studio
  A to studio B and four roads, say B1, B2, B3, and
  B4, from studio B to studio C. In how many ways
  can Morgan go from studio A to studio C and have
  lunch with Brad and Jennifer at Studio B?

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Basic Counting Principles

• There are 3 ways to go from studio A to studio B
  and 4 ways to go from studio B to studio C.
• The number of ways to go from studio A to studio
  C via studio B is 3 4 12.

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Basic Counting Principles
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Basic Counting Principles

• Consider two finite sets, X1 and X2. Then

• This is called the inclusion-exclusion principle
  for two finite sets.

• Consider three finite sets, A, B, and C. Then

• This is called the inclusion-exclusion principle
  for three finite sets.

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Basic Counting Principles
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Pigeonhole Principle

• The pigeonhole principle is also known as the
  Dirichlet drawer principle, or the shoebox
  principle.

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Pigeonhole Principle
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Pigeonhole Principle
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Permutations
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Permutations
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Combinations
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Combinations