ICS 253: Discrete Structures I

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ICS 253 Discrete Structures I
King Fahd University of Petroleum
Minerals Information Computer Science Department

• Counting and Applications

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Section 5.2 The Pigeonhole Principle

• Theorem If k is a positive integer and k 1 or
  more objects are placed into k boxes, then there
  is at least one box containing two or more of the
  objects.
• Proof
• Corollary A function f from a set with k 1 or
  more elements to a set with k elements is not
  one-to-one.

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Examples

• Example 1 How many people do you need to ensure
  that two of them have the same birthday (day and
  month)
• Example 2 How many English words do you need to
  ensure that two of them begin with the same
  letter.
• Example 3 How many students must be in a class
  to guarantee that at least two students receive
  the same score on the final exam, if the exam is
  graded on a scale from 0 to 100 points?

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The Generalized Pigeonhole Principle

• Theorem If N objects are placed into k boxes,
  then there is at least one box containing at
  least ?N/k? objects.

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Examples

• Example 1 What is the minimum number of students
  required in a discrete mathematics class to be
  sure that at least six will receive the same
  grade, if there are five possible grades, A, B,
  C, D, and F?

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Examples

• Example 2 How many cards must be selected from a
  standard deck of 52 cards to guarantee that at
  least
• three cards of the same suit are chosen?
• three hearts are selected?

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Examples

• Example 3 What is the least number of area codes
  needed to guarantee that the 25 million phones in
  a state can be assigned distinct 10-digit
  telephone numbers?
• Assume that telephone numbers are of the form
  NXX-NXX-XXXX, where the first three digits form
  the area code, N represents a digit from 2 to 9
  inclusive, and X represents any digit.

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Examples

• Example 4 Suppose that a computer science
  laboratory has 15 workstations and 10 servers. A
  cable can be used to directly connect a
  workstation to a server. For each server, only
  one direct connection to that server can be
  active at any time. We want to guarantee that at
  any time any set of 10 or fewer workstations can
  simultaneously access different servers via
  direct connections. what is the minimum number of
  direct connections needed to achieve this goal?
• Note that we could do this by connecting every
  workstation directly to every server (using 150
  connections),

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Examples

• Example 5 During a month with 30 days, a
  baseball team plays at least one game a day, but
  no more than 45 games. Show that there must be a
  period of some number of consecutive days during
  which the team must play exactly 14 games.

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Examples

• Example 6 Assume that in a group of six people,
  each pair of individuals consists of two friends
  or two enemies. Show that there are either three
  mutual friends or three mutual enemies in the
  group.

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Section 5.3 Permutations and Combinations

• Introduction
• Many counting problems can be solved by finding
  the number of ways to arrange a specified number
  of distinct elements of a set of a particular
  size, where the order of these elements matters.
• Many other counting problems can be solved by
  finding the number of ways to select a particular
  number of elements from a set of a particular
  size, where the order of the elements selected
  does not matter.
• In how many ways can we select three students
  from a group of five students to stand in line
  for a picture?
• How many different committees of three students
  can be formed from a group of four students?

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Permutations

• A permutation of a set of distinct objects is an
  ordered arrangement of these objects.
• An ordered arrangement of r elements of a set is
  called an r-permutation.
• Let S 1, 2, 3.
• An ordered permutation of S is
• A 2-permutation of S is
• The number of r-permutations of a set with n
  elements is denoted by P(n, r)
• Which we can compute using the product rule!

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P(n,r)

• If n is a positive integer and r is an integer
  with 1 ? r ? n, then, there are
  __________________ r-permutations of a set with n
  distinct elements.
• Note that P(n,0) 1.
• Corollary If n and r are integers with 0 ? r ? n
  then

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Examples

• Example 1 How many ways are there to select a
  first-prize winner, a second-prize winner, and a
  third-prize winner from 100 different people who
  have entered a contest?
• Example 2 Suppose that a saleswoman has to visit
  eight different cities. She must begin her trip
  in a specified city, but she can visit the other
  seven cities in any order she wishes. How many
  possible orders can the saleswoman use when
  visiting these cities?
• Example 3 How many permutations of the letters
  ABCDEFGH contain the string ABC?

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Combinations

• An r-combination of elements of a set is an
  unordered selection of r elements from the set.
• an r-combination is simply a subset of the set
  with r elements.
• The number of r-combinations of a set with n
  distinct elements is denoted by C(n, r).
• Note that C(n, r) is also denoted by and is
  called a binomial coefficient.

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Example

• What is C(4,2)?

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Important Results on Combinations

• Theorem The number of r-combinations of a set
  with n elements, where n is a nonnegative integer
  and r is an integer with 0 ? r ? n , equals
• Corollary Let n and r be nonnegative integers
  with r ? n. Then C(n, r) C (n, n r).

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Examples

• Example 1 How many ways are there to select five
  players from a 10-member tennis team to make a
  trip to a match at another school?
• Example 2 A group of 30 people have been trained
  as astronauts to go on the first mission to Mars.
  How many ways are there to select a crew of six
  people to go on this mission (assuming that all
  crew members have the same job)?

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Examples

• Example 3 How many bit strings of length n
  contain exactly r 1s?
• Example 4 Suppose that there are 9 faculty
  members in the mathematics department and 11 in
  the computer science department. How many ways
  are there to select a committee to develop a
  discrete mathematics course at a school if the
  committee is to consist of three faculty members
  from the mathematics department and four from the
  computer science department?