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Counting II

                               A.Benedict Balbuena
               Institute of Mathematics, University of the Philippines in Diliman



                                          15.1.2008




A.B.C.Balbuena (UP-Math)                   Counting II                              15.1.2008   1/9
Permutation


Definition
A permutation of a set of distinct elements is an ordered
arrangement (or sequence) of these elements. An ordered
arrangement of r elements of a set is called an r-permutation.

3-permutations of the set {a, b, c}:

              (a, b, c)(a, c, b)(b, a, c)(b, c, a)(c, a, b)(c, b, a)

How many permutations of an n-element set are there?



  A.B.C.Balbuena (UP-Math)           Counting II                       15.1.2008   2/9
Permutation


Definition
A permutation of a set of distinct elements is an ordered
arrangement (or sequence) of these elements. An ordered
arrangement of r elements of a set is called an r-permutation.

3-permutations of the set {a, b, c}:

              (a, b, c)(a, c, b)(b, a, c)(b, c, a)(c, a, b)(c, b, a)

How many permutations of an n-element set are there?



  A.B.C.Balbuena (UP-Math)           Counting II                       15.1.2008   2/9
Permutation


Definition
A permutation of a set of distinct elements is an ordered
arrangement (or sequence) of these elements. An ordered
arrangement of r elements of a set is called an r-permutation.

3-permutations of the set {a, b, c}:

              (a, b, c)(a, c, b)(b, a, c)(b, c, a)(c, a, b)(c, b, a)

How many permutations of an n-element set are there?



  A.B.C.Balbuena (UP-Math)           Counting II                       15.1.2008   2/9
Permutation Rule

Theorem
Given a set of n distinct elements, you wish to select r elements
from n and arrange them in r positions. The number of
                                                              n!
permutations of n elements taken r at a time is equal to
                                                           (n − r )!

In a race with eight runners, how many ways can the gold, silver and
bronze medals be awarded?
A salesperson has to travel eight cities but must begin the trip at a
specified city. He can visit the other seven cities in any order. How
many possible orders for the trip can the salesperson have?



  A.B.C.Balbuena (UP-Math)      Counting II                  15.1.2008   3/9
Permutation Rule

Theorem
Given a set of n distinct elements, you wish to select r elements
from n and arrange them in r positions. The number of
                                                              n!
permutations of n elements taken r at a time is equal to
                                                           (n − r )!

In a race with eight runners, how many ways can the gold, silver and
bronze medals be awarded?
A salesperson has to travel eight cities but must begin the trip at a
specified city. He can visit the other seven cities in any order. How
many possible orders for the trip can the salesperson have?



  A.B.C.Balbuena (UP-Math)      Counting II                  15.1.2008   3/9
Partition Rule

Theorem
The number of permutations of n elements, where there are n1
objects of type 1, n2 objects of type 2, ..., and nk objects of type
           n!
k, is
      n1 !n2 !...nk !
note: n1 + n2 + ... + nk = n
  1   How many ways are there to rearrange the letters in the word
      BOOKKEEPER?
  2   How many n-bit sequences contain exactly k ones?



  A.B.C.Balbuena (UP-Math)      Counting II                 15.1.2008   4/9
Combinations

Definition
A combination of a set is an unordered selection the set’s
elements. An unordered selection of r elements of a set is
called an r-combination.
An r-combination can be interpreted as a subset of the set with r
elements.
 1   In how many ways can I select 5 books from my collection of 100
     to bring on vacation?
 2   What is the number of k -element subsets of an n-element set?



  A.B.C.Balbuena (UP-Math)     Counting II                 15.1.2008   5/9
Combinations

Definition
A combination of a set is an unordered selection the set’s
elements. An unordered selection of r elements of a set is
called an r-combination.
An r-combination can be interpreted as a subset of the set with r
elements.
 1   In how many ways can I select 5 books from my collection of 100
     to bring on vacation?
 2   What is the number of k -element subsets of an n-element set?



  A.B.C.Balbuena (UP-Math)     Counting II                 15.1.2008   5/9
Theorem
Given a set of n distinct elements, you wish to select r elements
from n, then the number of combinations of r elements that can
                                                  n!
be selected from the n elements is equal to
                                             r !(n − r )!

total number of possible arrangements divided by number of
                                             n!
                                           (n−r )!
ways any one set can be rearranged =
                                             r!




  A.B.C.Balbuena (UP-Math)   Counting II                15.1.2008   6/9
Pigeonhole Principle


Theorem
Let k be an integer. If k + 1 or more objects are places into k
boxes, then there is at least one box containing two or more of
the objects.

Proof.
Suppose none of the k boxes contain more than one object.
Then the number of objects would be at most k . Contradiction,
since there are k + 1 objects.




  A.B.C.Balbuena (UP-Math)   Counting II                15.1.2008   7/9
Examples


Given two nonempty sets X , Y with |X | > |Y | , then for every
function f : X → Y there exist two different elements of X that
are mapped to the same element of Y .

 1   Among any group of 367 people, there must be at least two
     with the same birthday.
 2   How many students must be in a class to guarantee that at
     least two students have the same score, if the exam is
     graded from 0 to 100 and no half points allowed?




  A.B.C.Balbuena (UP-Math)    Counting II               15.1.2008   8/9
Examples


Given two nonempty sets X , Y with |X | > |Y | , then for every
function f : X → Y there exist two different elements of X that
are mapped to the same element of Y .

 1   Among any group of 367 people, there must be at least two
     with the same birthday.
 2   How many students must be in a class to guarantee that at
     least two students have the same score, if the exam is
     graded from 0 to 100 and no half points allowed?




  A.B.C.Balbuena (UP-Math)    Counting II               15.1.2008   8/9
Generalized Pigeonhole Principle



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


How many non-bald people in Metro Manila have the same
number of hairs on their head?




  A.B.C.Balbuena (UP-Math)    Counting II                15.1.2008   9/9

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counting

  • 1. Counting II A.Benedict Balbuena Institute of Mathematics, University of the Philippines in Diliman 15.1.2008 A.B.C.Balbuena (UP-Math) Counting II 15.1.2008 1/9
  • 2. Permutation Definition A permutation of a set of distinct elements is an ordered arrangement (or sequence) of these elements. An ordered arrangement of r elements of a set is called an r-permutation. 3-permutations of the set {a, b, c}: (a, b, c)(a, c, b)(b, a, c)(b, c, a)(c, a, b)(c, b, a) How many permutations of an n-element set are there? A.B.C.Balbuena (UP-Math) Counting II 15.1.2008 2/9
  • 3. Permutation Definition A permutation of a set of distinct elements is an ordered arrangement (or sequence) of these elements. An ordered arrangement of r elements of a set is called an r-permutation. 3-permutations of the set {a, b, c}: (a, b, c)(a, c, b)(b, a, c)(b, c, a)(c, a, b)(c, b, a) How many permutations of an n-element set are there? A.B.C.Balbuena (UP-Math) Counting II 15.1.2008 2/9
  • 4. Permutation Definition A permutation of a set of distinct elements is an ordered arrangement (or sequence) of these elements. An ordered arrangement of r elements of a set is called an r-permutation. 3-permutations of the set {a, b, c}: (a, b, c)(a, c, b)(b, a, c)(b, c, a)(c, a, b)(c, b, a) How many permutations of an n-element set are there? A.B.C.Balbuena (UP-Math) Counting II 15.1.2008 2/9
  • 5. Permutation Rule Theorem Given a set of n distinct elements, you wish to select r elements from n and arrange them in r positions. The number of n! permutations of n elements taken r at a time is equal to (n − r )! In a race with eight runners, how many ways can the gold, silver and bronze medals be awarded? A salesperson has to travel eight cities but must begin the trip at a specified city. He can visit the other seven cities in any order. How many possible orders for the trip can the salesperson have? A.B.C.Balbuena (UP-Math) Counting II 15.1.2008 3/9
  • 6. Permutation Rule Theorem Given a set of n distinct elements, you wish to select r elements from n and arrange them in r positions. The number of n! permutations of n elements taken r at a time is equal to (n − r )! In a race with eight runners, how many ways can the gold, silver and bronze medals be awarded? A salesperson has to travel eight cities but must begin the trip at a specified city. He can visit the other seven cities in any order. How many possible orders for the trip can the salesperson have? A.B.C.Balbuena (UP-Math) Counting II 15.1.2008 3/9
  • 7. Partition Rule Theorem The number of permutations of n elements, where there are n1 objects of type 1, n2 objects of type 2, ..., and nk objects of type n! k, is n1 !n2 !...nk ! note: n1 + n2 + ... + nk = n 1 How many ways are there to rearrange the letters in the word BOOKKEEPER? 2 How many n-bit sequences contain exactly k ones? A.B.C.Balbuena (UP-Math) Counting II 15.1.2008 4/9
  • 8. Combinations Definition A combination of a set is an unordered selection the set’s elements. An unordered selection of r elements of a set is called an r-combination. An r-combination can be interpreted as a subset of the set with r elements. 1 In how many ways can I select 5 books from my collection of 100 to bring on vacation? 2 What is the number of k -element subsets of an n-element set? A.B.C.Balbuena (UP-Math) Counting II 15.1.2008 5/9
  • 9. Combinations Definition A combination of a set is an unordered selection the set’s elements. An unordered selection of r elements of a set is called an r-combination. An r-combination can be interpreted as a subset of the set with r elements. 1 In how many ways can I select 5 books from my collection of 100 to bring on vacation? 2 What is the number of k -element subsets of an n-element set? A.B.C.Balbuena (UP-Math) Counting II 15.1.2008 5/9
  • 10. Theorem Given a set of n distinct elements, you wish to select r elements from n, then the number of combinations of r elements that can n! be selected from the n elements is equal to r !(n − r )! total number of possible arrangements divided by number of n! (n−r )! ways any one set can be rearranged = r! A.B.C.Balbuena (UP-Math) Counting II 15.1.2008 6/9
  • 11. Pigeonhole Principle Theorem Let k be an integer. If k + 1 or more objects are places into k boxes, then there is at least one box containing two or more of the objects. Proof. Suppose none of the k boxes contain more than one object. Then the number of objects would be at most k . Contradiction, since there are k + 1 objects. A.B.C.Balbuena (UP-Math) Counting II 15.1.2008 7/9
  • 12. Examples Given two nonempty sets X , Y with |X | > |Y | , then for every function f : X → Y there exist two different elements of X that are mapped to the same element of Y . 1 Among any group of 367 people, there must be at least two with the same birthday. 2 How many students must be in a class to guarantee that at least two students have the same score, if the exam is graded from 0 to 100 and no half points allowed? A.B.C.Balbuena (UP-Math) Counting II 15.1.2008 8/9
  • 13. Examples Given two nonempty sets X , Y with |X | > |Y | , then for every function f : X → Y there exist two different elements of X that are mapped to the same element of Y . 1 Among any group of 367 people, there must be at least two with the same birthday. 2 How many students must be in a class to guarantee that at least two students have the same score, if the exam is graded from 0 to 100 and no half points allowed? A.B.C.Balbuena (UP-Math) Counting II 15.1.2008 8/9
  • 14. Generalized Pigeonhole Principle Theorem If N objects are placed into k boxes, then there is at least one box containing at least N objects. k How many non-bald people in Metro Manila have the same number of hairs on their head? A.B.C.Balbuena (UP-Math) Counting II 15.1.2008 9/9