Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our User Agreement and Privacy Policy.

Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our Privacy Policy and User Agreement for details.

Successfully reported this slideshow.

Like this presentation? Why not share!

1,501 views

Published on

No Downloads

Total views

1,501

On SlideShare

0

From Embeds

0

Number of Embeds

23

Shares

0

Downloads

53

Comments

0

Likes

2

No embeds

No notes for slide

- 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

No public clipboards found for this slide

Be the first to comment