Transcript CMSC 203 / 0201 Fall 2002 – 14/16 October 2002 Week #8

CMSC 203 / 0201
Fall 2002
Week #8 – 14/16 October 2002
Prof. Marie desJardins
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TOPICS
 Counting
 Inclusion-exclusion
 Tree diagrams
 Pigeonhole principle
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MON 10/14
COUNTING 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)
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WED 10/16
PIGEONHOLE 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.
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FRI 10/11
** NO CLASS **
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