Chapter 6: Counting Methods and the Pigeonhole Principle

1
Chapter 6 Counting Methods and the Pigeonhole
Principle

• Discrete Mathematics
• ???

2
Multiplication Principle

• If an activity can be structured in t
• successive steps and step 1 can be done in
• n1 ways, step 2 can be done in n2 ways,,
• and step t can be done in nt ways, then the
• number of different possible activities is

3
Example 6.1.3

• How many strings of length 4 can be formed using
  the letters ABCDE if repetitions are not allowed?
• How many strings of part (a) begin with the
  letter B?
• How many strings of part (a) do not begin with
  the letter B?

4
Addition Principle

• Suppose that are sets and that
• the ith set Xi has ni elements. If X1,,Xt
• is a pairwise disjoint family (if
  ),
• the number of possible elements that can be
• selected from X1 or X2or Xt is

5
Example 6.1.8

• In how many ways can we select two books
• from different subjects among five distinct
• computer science, three distinct
• mathematics, and two distinct art books?

6
Example 6.1.9

• A six-person committee composed of Alice, Ben,
• Connie, Dolph, Egbert, and Francisco is to select
  a
• chairman, secretary, and treasurer?
• In how many ways can this be done?
• In how many ways can this be done if either Alice
  or Ben must be chairman?
• In how many ways can this be done if Egbert must
  hold one of the officers?
• In how many ways can this be done if both Dolph
  and Francisco must hold office?

7
Permutation

• A permutation of n distinct elements
• x1,,xn is an ordering of the n elements
• x1,,xn
• Theorem 6.2.3
• There are n! permutations of n elements.

8
r-permutation

• An r-permutation of n (distinct) elements of
• x1,,xn is an ordering of an r-element
• subset of x1,,xn. The number of
• r-permutations of a set of n distinct elements
• is denoted P(n,r)
• P(n,r)n(n-1)(n-2)(n-r1),

9
Example 6.2.12

• In how many ways can we select a
• chairperson, vice-chairperson, secretary,
• and treasurer from a group of 10 persons?

10
Example 6.2.14

• In how many ways can seven Martians and
• five Jovians wait in line if no two Jovians
• stand together?

11
Combination

• Given a set containing n
  (distinct)
• elements.
• An r-combination of X is an unordered selection
  of r-element of X
• The number of r-combinations of a set of n
  distinct elements is denoted

12
Example 6.2.19

• In how many ways can we select a
• committee of two women and three men
• from a group of five women and six men?

13
Theorem 6.6.2

• Suppose that a sequence S of n items has
• n1 identical objectives of type 1, n2
• identical objectives of type 2, , and nt
• identical objectives of type t. Then the
• number of orderings of S is

14
Example 6.6.3

• In how many ways can eight distinct books
• be divided among three students if Bill gets
• four books and Shizuo and Marian each get
• two books?

15
Example 6.6.4

• Consider three books a computer science
• book, a physics book, and a history book.
• Suppose that the library has at least six
• copy of each of these books. In how many
• ways can we select six book?

16
Theorem 6.6.5

• If X is a set containing t elements, the
• number of unordered, k-element selections
• from X, repetitions allowed, is

17
Example 6.6.6

• Suppose that there are plies of red, blue,
• and green balls and that each pile contains
• at least eight balls.
• In how many ways can we select eight
• balls?
• (b) In how many ways can we select eight balls if
  we must have at least one ball of each color?

18
Pigeonhole Principle (First Form)

• If n pigeons fly into k pigeonholes and k lt n,
• some pigeonhole contains at least two
• pigeons.

19
Example 1

• If the score of discrete mathematics ranged
• between 0 and 100, then, how many
• students are required to guarantee that
• there are at least two students have the
• same score?

20
Pigeonhole Principle (Second Form)

• If f is a function from a finite set X to a
  finite
• set Y and X gt Y, then f(x1) f(x2) for some

21
Example 6.8.3

• Show that if we select 151 distinct computer
• science courses numbered between 1 and
• 300 inclusive, at least two are consecutively
• numbered.

22
Example 6.8.4

• An inventory consists of a list of 80 items,
• each marked available or unavailable.
• There are 45 available items. Show that
• there are at least two available items in the
• list exactly nine items apart.

23
Pigeonhole Principle (Third Form)

• Let f be a function from a finite set X into a
• finite set Y. Suppose that X n and Y
• m. Let k n / m. Then there are at least k
• values a1, a2,,ak X such that

24
Example 2

• Show that at least how many people in a
• 100 person group are born in the same month?
• (b) In some month with 30 days, the baseball team
  A has at least one game played in each day, but
  the total games played in this month will not
  exceeded 45. Prove that the team A exactly played
  14 games in a consecutively period of days.

25
Example 6.8.5

• A useful feature of black-and-white pictures
• is the average brightness of the picture. Let
• us say that two pictures are similar if their
• average brightness differs by no more than
• some fixed value. Show that among six
• pictures, there are either three that are
• mutually similar or three that are mutually
• dissimilar.