Fall 2009
CS131 Combinatorial Structures
Homework 2
Homework 2, due Sept 22
You must prove your answer to every question. Do not rely only on the homework for exercise: there are several self-check exercises of the easier kind in the book, try to solve th
Discrete Mathematics
Dr. Fred Phelps
Lecture 1
Counting and
Set Theory
Announcements
Please register for discrete math on the
site http:/dl.iitu.kz. The main textbook is
on that site.
The syllabus is also on that site.
And a copy of the main textbook:
Dis
Discrete Mathematics
Dr. Fred Phelps
Lecture 2
More Counting:
Permutations and
Combinations
Announcements
Please register for discrete math on the
site http:/dl.iitu.kz. The main textbook is
on that site.
The syllabus is also on that site.
And a copy of t
Discrete Mathematics
Dr. Fred Phelps
Lecture 3
Induction and
Stirlings
Formula
LVP 2.1 Mathematical
Induction n
A tool to prove formulae for every .
What is the sum of the first n odd numbers,
n
i.e.
(2i 1)?
i =1
Lets experiment:
n
Conjecture:
(2i 1) = n
Discrete Mathematics
Dr. Fred Phelps
Lecture 4
The Pigeonhole
Principle
& The Binomial
Theorem
LVP 2.4 Pigeonhole Principle
If we have n boxes and we
place more than n objects
into them, then there will be
at least one box that
contains more than one
obje
Discrete Mathematics
Dr. Fred Phelps
Lecture 5
The InclusionExclusion Principle
& More Counting
Problems
LPV 2.3 & Chen 13 The
Inclusion-Exclusion Principle
1.
This is covered in LPV 2.3 but the lecture
is taken from Chens book.
Consider the sets S = cfw_
Discrete Mathematics
Dr. Fred Phelps
Lecture 6
Pascals
Triangle and the
Normal
LPV 3.5 Pascals Triangle
nth row consists
n n n n
, ,.
, .
0 1 n 1 n
LPV 3.6 Pascals Triangle identities
Recall the identities we
proved earlier:
1)
The triangle is
symmetr
Combinatorial Structures
Freely using the textbook by Lovsz-Pelikn-Vesztergombi
Pter Gcs
Computer Science Department
Boston University
Fall 2009
Pter Gcs (Boston University)
CS 131
Fall 09
1 / 176
Introduction
Introduction
For details on the course struct
Week Five Problems Recurrence Relations
Name 1_
Name 2_
Name 3_
Group_
Work in groups of three (or two if necessary). Please write the solutions nicely on the back of this
paper and continue on more paper not tetrads.
1) For each of these tell the degree
Week One Solutions
1) Consider a group of five men and four women with the following hand shaking rules and
calculate how many possible handshakes there are:
Only men shake hands with each other. How many possible handshakes are there?
Handshakes are bet
Week One SSWT Assignment
1) Consider a group of five men and four women with the following hand shaking rules and
calculate how many possible handshakes there are:
Only men shake hands with each other. How many possible handshakes are there?
Men shake h
Fall 2009
CS131 Combinatorial Structures
Self-test questions
Self-test questions
These questions are provided for additional self-testing, on material for which the book does not have enough self-test questions. Problem 1. Let A = cfw_a, b, c, B = cfw_ b,
Fall 2009
CS131 Combinatorial Structures
Midterm exam 2
Midterm exam 2
Only a single hand-written crib sheet can be used, no books or notes. Even if I ask
for just a yes/no answer, you must always give a proof. You may get some points
even if you write I
Fall 2009
CS131 Combinatorial Structures
Homework 3
Homework 3, due Sept 29
You must prove your answer to every question. Do not rely only on the homework for exercise: there are several self-check exercises of the easier kind in the book, try to solve th
Fall 2009
CS131 Combinatorial Structures
Homework 1
Homework 1, due Sept 15
You must prove your answer to every question. Do not rely only on the homework for exercise: there are several self-check exercises of the easier kind in the book, try to solve th
Fall 2009
CS131 Combinatorial Structures
Homework 4
Homework 4, due Oct 15
You must prove your answer to every question. Do not rely only on the homework for exercise: there are several self-check exercises of the easier kind in the book, try to solve the
Fall 2009
CS131 Combinatorial Structures
Homework 5
Homework 5, due Oct 20
You must prove your answer to every question. Do not rely only on the homework for exercise: there are several self-check exercises of the easier kind in the book, try to solve the
Fall 2009
CS131 Combinatorial Structures
Homework 6
Homework 6, due Oct 27
You must prove your answer to every question.
Do not rely only on the homework for exercise: there are several self-check exercises of the easier kind in the book, try to solve the
Fall 2009
CS131 Combinatorial Structures
Homework 7
Homework 7, due Nov 3
You must prove your answer to every question.
Do not rely only on the homework for exercise: there are several self-check exercises of the easier kind in the book, try to solve them
Fall 2009
CS131 Combinatorial Structures
Homework 8
Homework 8, due Nov 17
You must prove your answer to every question. Do not rely only on the homework for exercise: there are several self-check exercises of the easier kind in the book, try to solve the
Fall 2009
CS131 Combinatorial Structures
Homework 9
Homework 9, due Nov 24
You must prove your answer to every question. Do not rely only on the homework for exercise: there are several self-check exercises of the easier kind in the book, try to solve the
Fall 2009
CS131 Combinatorial Structures
Homework 10
Homework 10, due Dec 8
You must prove your answer to every question.
Do not rely only on the homework for exercise: there are several self-check exercises of the easier kind in the book, try to solve th
Fall 2009
CS131 Combinatorial Structures
Midterm exam 1
Midterm exam 1
Only a single hand-written crib sheet can be used, no books or notes. Even if I ask for just a yes/no answer, you must always give a proof. You may get some points even if you write I
Second Week Practice Assignment
Counting Review:
1.8.27 Alice has 10 balls (all different). First, she splits them into two piles; then she picks one of
the piles with at least two elements, and splits it into two; she repeats this until each pile has onl