MAD 4203 and MAT 5932 Introduction to Combinatorics
Homework Solutions
Comments Encouraged
Chapter 3.
3.25 By the Rule of Product, there are 9 10 10 three digit integers xyz. Indeed, the first digit cant
be zero, so there are only 9 choices for x, but the
MAD 4203 and MAT 5932 Introduction to Combinatorics
Homework Solutions
Comments Encouraged
Chapter 3.
3.25 By the Rule of Product, there are 9 10 10 three digit integers xyz. Indeed, the first digit cant
be zero, so there are only 9 choices for x, but the
MAD 4203 and MAT 5932 Introduction to Combinatorics
Homework Solutions
Comments Encouraged
Chapter 3.
3.25 By the Rule of Product, there are 9 10 10 three digit integers xyz. Indeed, the first digit cant
be zero, so there are only 9 choices for x, but the
MAD 4203 and MAT 5932, Introduction to Combinatorics
Exam 3 Study Guide
Since Exam 2, we introduced the powerful methods of thePrinciple of Inclusion-Exclusion and of Generating Functions. While these constitute only two methods, weve seen that handling t
MAD 4203 and MAT 5932, Introduction to Combinatorics
Exam 2 Study Guide
Balls and Urns (so far)
Since Exam 1, weve addressed seven Ball and Urn distribution problems. Be able to categorize, describe
and evaluate these seven distributions for m balls and n
MAD 4932 and MAT 5932, Introduction to Combinatorics
Some Sums, Some More
1. Determine the number of integer solutions to
x1 + x2 + x3 + x4 = 19,
where 0 x1 5, 0 x2 6, 3 x3 7, 3 x4 8, by using
(a) inclusion-exclusion;
(b) generating functions.
2. Find a c
MAD 4203 and MAT 5932, Introduction to Combinatorics
Some Sums
1. Consider the equation
x1 + x2 + x3 + x4 = 100.
(a) How many positive integer solutions does this equation have?
(b) How many integer solutions does it have where x1 , x2 3 and x3 , x4 5?
(c
MAD 4203 and MAT 5932, Introduction to Combinatorics
Poker Problems
A poker hand consists of 5 cards from a standard deck (of 52), where we assume no cards are wild. How
many poker hands are . . .
1. flushes?
2. straights?
3. straight flushes?
4. four of
MAT 4930 Introduction to Graph Theory
(Alleged) Homework Solutions
Comments Encouraged
Introductory Problems from 9
9.23 Lets start with a warm-up. First, note
that there are n2 pairs 1 i < j n from the set
n
[n] = cfw_1, . . . , n. As such, there are 2 2
MAT 4930, Introduction to Graph Theory
Exam 3 Study Guide
Exam 3 will cover one concept from Chapter 11 (see below), and otherwise, will cover Chapter 12. The
exam will cover some homework from Chapter 12 (see below), and some material from lectures (see
MAT 4930, Introduction to Graph Theory
Exam 2 Study Guide
Exam 2 will cover some of Chapter 9, and some of Chapter 11. The exam will cover some homework
from these chapters (see below), and some material from lectures (see below).
Featured Homework Proble
MAT 4930, Introduction to Graph Theory
Exam 1 Study Guide
Exam 1 will cover some of Chapter 9, and most of Chapter 10. The exam will cover some homework
from these chapters (see below), and some material from lectures (see below).
Featured Homework Proble
1.
2.
3.
4.
5.
Convert the decimal 1023 to binary.
True or False, 37 is congruent to 3 modulo 7
What is the quotient and remainder when 777 is divided by 21?
Convert BADFACED from hexadecimal expansion to its binary expansion.
True or False, 93 is a prime
1. Suppose m and t are the propostions:
m: You are a member of the team
t: You take afternoon classes
express in English the compound proposition
.
2. Suppose d is the proposition "it is dark outside" and h is the proposition "I stay at home".
Express in
1. How many strings of all the first seven letters of the alphabet (A, B, C, D, E, F, G) are there that
contain no repeated letters and begin or end with a vowel (A or E).
2. Assume that you have an ordinary deck of 52 playing cards. How many possible 7-c
1. Find an inverse for a modulo m when a=2 and m=17, both are relatively prime integers using the
Euclidean algorithm.
2. Which memory location is assigned by the hashing function h(k)=k mod 101 to the record of
insurance company customer with the social
ExerciseSet1.3:p.21119odd
1a) No, Yes, No, Yes
b) R = cfw_(2,6), (2,8), (2,10), (3,6), (4,8)
c) Domain of
d)
R
2
R= A= cfw_2,3,4, co-domain of R = B = cfw_6,8,10]
6
3
8
4
10
3a) 3 T 0 because = 1, is an integer;
1 T (-1) because = 2/3, is not an integer;
Assignment#1
ExerciseSet1.1:p.5,113odd
1a) x^2= -1
b) a real number x
3a) in between a and b
b) real numbers a and b, theres a real number c
5a) r is positive
b) positive, 1/r is positive
c) is positive, 1/r is positive
7a) True; There are real numbers wh
MAD 5305 Graph Theory, Spring 2012
Classes: PHY 109, MWF 10:45-11:35am
CRN: 21139
Assignment 1 solution
Question 1.
(1 point) Exercise 1.1.9. Prove that the graph on the left is isomorphic to the
complement of the graph on the right.
1
a
1
0
1
0
2
5
6
1
0
MAD 5305 Graph Theory, Spring 2012
Classes: PHY 109, MWF 10:45-11:35am
CRN: 21139
Assignment 3 solution
Question 1. (2 points) Exercise 1.4.29. Suppose that G is a graph and D is a strong orientation
of G.
(a) Consider an edge cfw_x, y in G. Prove that i
MAD 5305 Graph Theory, Spring 2012
Classes: PHY 109, MWF 10:45-11:35am
CRN: 21139
Assignment 2 solution
Question 1.
(2 points) (a) A graph G is dened to be connected if every pair of vertices is
connected by a path. Let G be a graph and x V (G). Prove tha
MAD 5305 Graph Theory, Spring 2012
Classes: PHY 109, MWF 10:45-11:35am
CRN: 21139
Assignment 4 solution
Question 1. (2 points) (a) Describe the breadth rst search (BFS) algorithm when applied to a
connected graph.
(b) Explain how BFS can be used to nd the
MAD 5305 Graph Theory, Spring 2012
Classes: PHY 109, MWF 10:45-11:35am
CRN: 21139
Assignment 5 solution
Question 1.
(1 point) Exercise 3.1.8. Prove that every tree T contains at most one perfect
matching. (Hint. If M and M are two perfect matchings in T ,
MAD 5305 Graph Theory, Spring 2012
Classes: PHY 109, MWF 10:45-11:35am
CRN: 21139
Assignment 8 solution
Question 1. Exercises 7.2.4 and 7.2.7. Note that a Hamiltonian path in a graph G is a spanning
path of G.
(a) Prove that if G has a Hamiltonian path th
MAD 5305 Graph Theory, Spring 2012
Classes: PHY 109, MWF 10:45-11:35am
CRN: 21139
Assignment 7 solution
Question 1. (1 point) Exercise 5.1.12, 5.1.13, 5.1.14, 5.1.15.
(a) Let (G) denote the independence number of of a graph G. Prove or disprove: If G is a
MAD 5305 Graph Theory, Spring 2012
Classes: PHY 109, MWF 10:45-11:35am
CRN: 21139
Assignment 6 solution
Question 1. (2 points) Exercise 3.3.16. Let G be a k -regular graph of even order that remains
connected when any k 2 edges are removed. We shall follo
MAD 6206 Combinatorics I, Fall 2011
MWF 12:55-1:45pm, PHY 120
Instructor:
Oce:
Email:
Oce hours:
CRN #: 87401
Stephen Suen
PHY356
[email protected]
T 9-10am, R 8:30-10:30am (to be conrmed).
Course Description We shall consider enumeration techniques in this c