September1999

CMSC 203 / 0201Fall 2002

Week #8 – 14/16 October 2002

Prof. Marie desJardins

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TOPICS

Counting Inclusion-exclusion Tree diagrams Pigeonhole principle

September1999

MON 10/14COUNTING BASICS (4.1)

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Concepts/Vocabulary

Counting Sum rule |A1 A2 … Am| = |A1| + … + |Am| for

disjoint Ai

Product rule |A1 x A2 x … x Am| = |A1| |A2| … |Am|

Inclusion-exclusion |A1 A2| = |A1| + |A2| - |A1 A2|

Tree diagrams

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Examples

Exercise 4.1.3: A multiple-choice test contains 10 questions. There are four possible answers for each question. (a) How many ways can a student answer the questions

on the test if every question is answered? (b) How many ways can a student answer the questions

on the test if the student can leave answers blank?

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Examples II

How many bit strings of length 8 are there? How many bit strings of length 8 or less are there? How many bit strings of length 8 or more are

there? ☺ Exercise 4.1.13: How many bit strings with length

not exceeding n, where n is a positive integer? How many such strings consisting entirely of 1s?

How many functions are there from A B where |A| = m and |B| = n? How many 1-to-1 functions are there?

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Examples III

In how many ways can six elements a1…a6 be placed into an array if: (a) a1 and a2 must be in adjacent positions (not

necessarily in that order) (b) a1 and a2 must not be in adjacent positions

(c) a1 must have a lower index than a6

(Analogous to Exercise 4.1.39.)

Exercise 4.1.50: Use a tree diagram to find the number of ways that the World Series can occur (four games out of seven wins the series)

September1999

WED 10/16PIGEONHOLE PRINCIPLE (3.5)

** HOMEWORK #5 DUE **

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Concepts / Vocabulary

Pigeonhole Principle If k+1 or more objects are in k boxes, at least one box

has two or more objects

Generalized pigeonhole principle If N objects are in k boxes, one box has at least

N/k objects

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Examples

Exercise 4.2.4: A bowl contains 10 red balls and 10 blue balls. A woman selects balls at random without looking at them. (a) How many balls must she select to be sure of having

at least three balls of the same color? (b) How many balls must she select to be sure of having

at lets three blue balls? Exercise 4.2.9: How many students, each of whom

comes from one of the 50 states, must be enrolled in a university to guarantee that there are at lets 100 who come from the same state?

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Examples II

Example 4.2.10 (page 248): Assume that in a group of six people, each pair of individuals consists of two friends or two enemies. Show that there are either three mutual friends or three mutual enemies in the group.

Exercise 4.2.29: A computer network consists of six computers. Each computer is directly connected to zero or more of the other computers. Show that there are at least two computers in the network that are directly connected to the same number of other computers.

September1999

FRI 10/11** NO CLASS **