Chapter 7: Counting Principles

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Chapter 7 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

• Explore generalized permutations and combinations
• 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
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Generalized Permutations and Combinations
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Generalized Permutations and Combinations
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Binomial Coefficients

• The expression x y is a binomial expression as
  it is the sum of two terms.
• The expression (x y)n is called a binomial
  expression of order n.

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Binomial Coefficients
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Binomial Coefficients
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Binomial Coefficients

• Pascals Triangle
• The number C(n, r) can be obtained by
  constructing a triangular array.
• The row 0, i.e., the first row of the triangle,
  contains the single entry 1. The row 1, i.e., the
  second row, contains a pair of entries each equal
  to 1.
• Calculate the nth row of the triangle from the
  preceding row by the following rules

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Binomial Coefficients
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Binomial Coefficients

• ALGORITHM 7.1 Determine the factorial of a
  nonnegative integer.
• Input na positive integer
• Output n!
• function factorial(n)
• begin
• fact 1
• for i 2 to n do
• fact fact i
• return fact
• end

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Binomial Coefficients

• The technique known as divide and conquer can be
  used to compute C(n, r ).
• In the divide-and-conquer technique, a problem is
  divided into a fixed number, say k, of smaller
  problems of the same kind.
• Typically, k 2. Each of the smaller problems is
  then divided into k smaller problems of the same
  kind, and so on, until the smaller problem is
  reduced to a case in which the solution is easily
  obtained.
• The solutions of the smaller problems are then
  put together to obtain the solution of the
  original problem.

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Binomial Coefficients
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Binomial Coefficients

• ALGORITHM 7.3 Determine C(n, r) using dynamic
  programming.
• Input n, r , n gt 0, r gt 0, r n
• Output C(n, r)
• function combDynamicProg(n,r)
• begin
• for i 0 to n do
• for j 0 to min(i,r) do
• if j 0 or j i then
• Ci,j 1
• else
• Ci,j Ci-1, j-1
  Ci-1, j
• return Cn, r
• end

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Generating Permutations and Combinations
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Generating Permutations and Combinations
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Generating Permutations and Combinations
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Generating Permutations and Combinations
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Generating Permutations and Combinations
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Generating Permutations and Combinations
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Discrete Probability

• Definition 7.8.1
• A probabilistic experiment, or random experiment,
  or simply an experiment, is the process by which
  an observation is made.
• In probability theory, any action or process that
  leads to an observation is referred to as an
  experiment.
• Examples include
• Tossing a pair of fair coins.
• Throwing a balanced die.
• Counting cars that drive past a toll booth.

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Discrete Probability

• Definition 7.8.3
• The sample space associated with a probabilistic
  experiment is the set consisting of all possible
  outcomes of the experiment and is denoted by S.
• The elements of the sample space are referred to
  as sample points.
• A discrete sample space is one that contains
  either a finite or a countable number of distinct
  sample points.

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Discrete Probability

• Definition 7.8.6
• An event in a discrete sample space S is a
  collection of sample points, i.e., any subset of
  S. In other words, an event is a set consisting
  of possible outcomes of the experiment.
• Definition 7.8.7
• A simple event is an event that cannot be
  decomposed. Each simple event corresponds to one
  and only one sample point. Any event that can be
  decomposed into more than one simple event is
  called a compound event.

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Discrete Probability

• Definition 7.8.8
• Let A be an event connected with a probabilistic
  experiment E and let S be the sample space of E.
  The event B of nonoccurrence of A is called the
  complementary event of A. This means that the
  subset B is the complement A of A in S.
• In an experiment, two or more events are said to
  be equally likely if, after taking into
  consideration all relevant evidences, none can be
  expected in reference to another.

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Discrete Probability
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Discrete Probability

• Axiomatic Approach
• Analyzing the concept of equally likely
  probability, we see that three conditions must
  hold.
• The probability of occurrence of any event must
  be greater than or equal to 0.
• The probability of the whole sample space must be
  1.
• If two events are mutually exclusive, the
  probability of their union is the sum of their
  respective probabilities.
• These three fundamental concepts form the basis
  of the definition of probability.

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Discrete Probability
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Discrete Probability
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Discrete Probability
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Discrete Probability

• Conditional Probability
• Consider the throw of two distinct balanced dice.
  To find the probability of getting a sum of 7,
  when it is given that the digit in the first die
  is greater than that in the second.
• In the probabilistic experiment of throwing two
  dice the sample space S consists of 6 6 36
  outcomes.
• Assume that each of these outcomes is equally
  likely. Let A be the event The sum of the digits
  of the two dice is 7, and let B be the event The
  digit in the first die is greater than the second.

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Discrete Probability

• Conditional Probability
• A (6, 1), (5 , 2), (4, 3), (3, 4), (2, 5), (1,
  6)
• B (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (5 ,
  1), (5 , 2), (5 , 3),(5 , 4), (4, 1), (4, 2), (4,
  3), (3, 1), (3, 2), (2, 1).
• Let C be the event The sum of the digits in the
  two dice is 7 but the digit in the first die is
  greater than the second. Then C (6, 1), (5 ,
  2), (4, 3) A n B.