Assignment 9 (not for credit)
1. Given a graph G, let G, the complement of G be dened as the graph with vertex set V (G),
and cfw_u, v is an edge in G if and only if cfw_u, v is not an edge in G. Let G be a graph with
p 11 vertices. Prove that either G
Assignment 8
Due: Friday, November 23, 12:00pm
Assignment drop box: Slot 9, CS Box 3 (pink box at 3rd oor MC), by bridge to DC
1. [Total Points: 7 ] Let G and H be graphs with vertex set cfw_1, 2, 3, . . . , 16 and edges
cfw_
E (G) = cfw_1, 3, cfw_1, 6, c
Assignment 7
Due: Friday, November 16, 12:00pm
Assignment drop box: Slot 9, CS Box 3 (pink box at 3rd oor MC), by bridge to DC
1. [Total Points: 7 ] Let An be the graph whose vertices are binary strings of length n and edges
are between strings that dier
Assignment 6
Due: Friday, November 2, 12:00pm
Assignment drop box: Slot 9, CS Box 3 (pink box at 3rd oor MC), by bridge to DC
1. [Total Points: 6 ] Solve the linear recurrence equation
bn + 4bn1 + 5bn2 + 2bn3 = 3n 5, n 3,
with initial conditions b0 = 3, b
Assignment 5
Due: Friday, October 26, 12:00pm
Assignment drop box: Slot 9, CS Box 3 (pink box at 3rd oor MC), by bridge to DC
1. [Total Points: 6 ] The Fibonacci sequence is a sequence of numbers starting with 0 and 1
and each subsequent number is the sum
Assignment 4
Due: Friday, October 19, 12:00pm
Assignment drop box: Slot 9, CS Box 3 (pink box at 3rd oor MC), by bridge to DC
1. [Total Points: 4 ] Express (x) by the summation of 1 (x) and 2 (x), where 1 (x) is a
polynomial and 2 (x) is a proper rational
Assignment 2
Due: Friday, September 28, 12:00pm
Assignment drop box: Slot 9, CS Box 3 (pink box at 3rd oor MC), by bridge to DC
1. [Total Points: 5 ]
(a) [Total Points: 3 ] Let S = cfw_0, 1, 2, . . . , 29. Let 1 be the weight function such that
1 ( ) equa
Assignment 1
Due: Friday, September 21, 12:00pm
Assignment drop box: Slot 9, CS Box 3 (pink box at 3rd oor MC), by bridge to DC
1. [Total Points: 3 ] How many binary strings of length 10 are there such that there are exactly
two 1s and they are not in con
A brief review
Elementary counting
Binomial coecients: counting the number of subsets of size k of a set with n
elements. (e.g. number of binary strings of length n which contains exactly k 1s.)
Sum and products: |A| = k=1 |Ai |, where Ai s are disjoint
More examples: combining all techniques
Problem 1: Let S1 be the set of all words formed using letters A, B and C that satisfy the following
condition.
* There are at most one A and two B s.
(For example, the ABC is a valid word, whereas BAA is NOT a word
More examples: combining all techniques
Problem 1: Let S1 be the set of all words formed using letters A, B and C that satisfy the following
condition.
* There are at most one A and two B s.
(For example, the ABC is a valid word, whereas BAA is NOT a word
Page 1
Math 229 Midterm Exam #2
3:35 5:15 (100 minutes), November 7, 2012
Name:
Signature:
Id.#:
Question
Value
1
9
2
9
3
10
4
12
Total
Mark Awarded
40
Instructions:
1. No notes, crib sheets, or calculators are allowed.
2. Show all work. For full credit a
Page 1
Math 229 Midterm Exam #1
3:35 5:15 (100 minutes), October 3, 2012
Name:
Signature:
Id.#:
Question
Value
1
9
2
11
3
7
4
13
Total
Mark Awarded
40
Instructions:
1. No notes, crib sheets, or calculators are allowed.
2. Show all work. For full credit al
Math 229: Introduction to Combinatorics
Fall 2012
Lecture: 01:30-02:20 MWF, MC 2034
Tutorial: 03:30-04:20 W, HH 139 (starting from the second week)
Instructor: Jane Pu Gao
Office hours: Mondays 2:30 3:30 pm
Email: p3gao@uwaterloo.ca
Thursdays 2:30 3:30 pm
Assignment 9 (not for credit)
1. Given a graph G, let G, the complement of G be dened as the graph with vertex set V (G),
and cfw_u, v is an edge in G if and only if cfw_u, v is not an edge in G. Let G be a graph with
p 11 vertices. Prove that either G
Assignment 8
Due: Wednesday, November 28, 12:00pm
Assignment drop box: Slot 9, CS Box 3 (pink box at 3rd oor MC), by bridge to DC
1. [Total Points: 7 ] Let G and H be graphs with vertex set cfw_1, 2, 3, . . . , 16 and edges
cfw_
E (G) = cfw_1, 3, cfw_1, 6
Assignment 7
Due: Friday, November 16, 12:00pm
Assignment drop box: Slot 9, CS Box 3 (pink box at 3rd oor MC), by bridge to DC
1. [Total Points: 7 ] Let An be the graph whose vertices are binary strings of length n and edges
are between strings that dier
Assignment 6
Due: Friday, November 2, 12:00pm
Assignment drop box: Slot 9, CS Box 3 (pink box at 3rd oor MC), by bridge to DC
1. [Total Points: 6 ] Solve the linear recurrence equation
bn + 4bn1 + 5bn2 + 2bn3 = 3n 5, n 3,
with initial conditions b0 = 3, b
Assignment 5
Due: Friday, October 26, 12:00pm
Assignment drop box: Slot 9, CS Box 3 (pink box at 3rd oor MC), by bridge to DC
1. [Total Points: 6 ] The Fibonacci sequence is a sequence of numbers starting with 0 and 1
and each subsequent number is the sum
Assignment 4
Due: Friday, October 19, 12:00pm
Assignment drop box: Slot 9, CS Box 3 (pink box at 3rd oor MC), by bridge to DC
1. [Total Points: 4 ] Express (x) by the summation of 1 (x) and 2 (x), where 1 (x) is a
polynomial and 2 (x) is a proper rational
Assignment 2
Due: Friday, September 28, 12:00pm
Assignment drop box: Slot 9, CS Box 3 (pink box at 3rd oor MC), by bridge to DC
1. [Total Points: 5 ]
(a) [Total Points: 3 ] Let S = cfw_0, 1, 2, . . . , 29. Let 1 be the weight function such that
1 ( ) equa
Assignment 1
Due: Friday, September 21, 12:00pm
Assignment drop box: Slot 9, CS Box 3 (pink box at 3rd oor MC), by bridge to DC
1. [Total Points: 3 ] How many binary strings of length 10 are there such that there are exactly
two 1s and they are not in con
Introduction to Graph Theory
What are graphs?
Denition 1 A graph G = (V, E ) where V is a nite nonempty set, which we call vertices,
and E is a set of unordered pairs of vertices in V , which we call edges.
Given a graph G, we use V (G) and E (G) to denot
Math 229: basic counting
1
What is counting?
Problem 1: Let there be a bag of n 3 distinct balls, labelled from 1 up to n. Now pick 3 balls
out of the bag sequentially. How many dierent ways can we do this?
Solution: There are n ways to pick the rst ball,
Applications of generating functions
Solutions to integer equations
Problem 1: Let k and n be xed integers. How many solutions are there to the equation
x1 + x2 + + xk = n,
where all xi are nonnegative integers?
For example, if n = 4 and k = 2, then there