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