[Hernandez 1213A] Cs30 day09

Counting

CS 30 : Discrete Mathematics for Computer Science First Semester, AY 2012-2013

https://sites.google.com/a/dcs.upd.edu.ph/nhsh_classes/cs 30

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Nestine Hope S. Hernandez Algorithms and Complexity Laboratory Department of Computer Science University of the Philippines, Diliman nshernandez@dcs.upd.edu.ph updilseal

Day 9

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acl-logo Discrete Mathematics for Computer Science

CS 30

Counting

Combinatorics and Computing

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1

Counting The Pigeonhole Principle

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acl-logo Discrete Mathematics for Computer Science

CS 30

Counting

The Pigeonhole Principle

QUIZ A particular brand of shirt comes in 12 colors, has a male version and a female version, and comes in three sizes for each sex. How many di erent types of this shirt are made?

1

2

3

How many di erent initials can someone have if a person has at least two, but no more than ve, di erent initials?

Assume that each initial is one of the 26 letters of the English alphabet.

How many strings of eight English letters are there a) if letters can be repeated? b) that start with X, if no letter can be repeated? c) that start and end with X, if letters can be repeated?

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acl-logo Discrete Mathematics for Computer Science

CS 30

Counting

The Pigeonhole Principle

The Pigeonhole Principle

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updilseal

dcs-logo

acl-logo Discrete Mathematics for Computer Science

CS 30

Counting

The Pigeonhole Principle

The Pigeonhole Principle

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updilseal

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acl-logo Discrete Mathematics for Computer Science

CS 30

Counting

The Pigeonhole Principle

The Pigeonhole Principle

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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

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objects. dcs-logo

acl-logo Discrete Mathematics for Computer Science

CS 30

Counting

The Pigeonhole Principle

The Pigeonhole Principle

Example 1

A function

f

from a set with

elements is not one-to-one.

2

k + 1 or more elements to a set with k

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Among any group of at least 13 people, there must be at least two with the same birthmonth.

3

Your bedroom drawer contains ten black and ten white socks. You reach in and pull some out without turning on the lights so as not to disturb your younger sibling who is sleeping in the same room. What is the least number of socks you must pull out to be sure to get a matched pair?

4

Show that for every integer

n there is a multiple of n that has both

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0s and 1s in its binary form.

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acl-logo Discrete Mathematics for Computer Science

CS 30

Counting

The Pigeonhole Principle

The Generalized Pigeonhole Principle

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updilseal

dcs-logo

acl-logo Discrete Mathematics for Computer Science

CS 30

Counting

The Pigeonhole Principle

The Generalized Pigeonhole Principle

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updilseal

dcs-logo

acl-logo Discrete Mathematics for Computer Science

CS 30

Counting

The Pigeonhole Principle

The Generalized Pigeonhole Principle

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If

N objects are placed into k boxes, then there is at least one box dN /k e objects.

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containing at least

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acl-logo Discrete Mathematics for Computer Science

CS 30

Counting

The Pigeonhole Principle

The Generalized Pigeonhole Principle

Example 1

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Among 100 people there are at least 9 who were born in the same month.

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acl-logo Discrete Mathematics for Computer Science

CS 30

Counting

The Pigeonhole Principle

The Generalized Pigeonhole Principle

Example 1

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Among 100 people there are at least 9 who were born in the same month.

2

(a) 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?

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dcs-logo

acl-logo Discrete Mathematics for Computer Science

CS 30

Counting

The Pigeonhole Principle

The Generalized Pigeonhole Principle

Example 1

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Among 100 people there are at least 9 who were born in the same month.

2

(a) 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? (b) How many cards must be selected from a standard deck of 52 cards to guarantee that at least three hearts are chosen? updilseal

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acl-logo Discrete Mathematics for Computer Science

CS 30

Counting

The Pigeonhole Principle

Some Elegant Applications of the Pigeonhole Principle

Example 1

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

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must play exactly 14 games.

2

Show that among any

n + 1 positive integers not exceeding 2n there

must be an integer that divides one of the other integers.

3

Every sequence of

n

2

+ 1 distinct real numbers contains a n + 1 that is either strictly increasing

subsequence of length strictly decreasing.

4

or

Assume that in a group of six people, each pair of individuals

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consists of two friends or two enemies. Show that there are either three mutual friends or three mutual enemies in the group. dcs-logo

acl-logo Discrete Mathematics for Computer Science

CS 30

Counting

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Questions?

See you next meeting!

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acl-logo Discrete Mathematics for Computer Science

CS 30