Math 239 Fall 2014 Assignment 9 Solutions
1. cfw_6 marks Let G be a 4-regular connected planar graph with an embedding where every face has degree 3 or 4,
and adjacent faces have dierent face degrees. Determine the number of vertices, edges, faces of degr
Math 239 Fall 2014 Assignment 10 Solutions
1. (a) cfw_4 marks Let d 2 N and let G be a graph where every vertex has degree at most d. Prove that G is
(d + 1)-colourable.
Solution. We prove by induction on the number of vertices n. When n = 1, the single v
Math 239 Winter 2015 Assignment 2 Solutions
1. cfw_4 marks Using mathematical induction on k, prove that for any integer k 1,
n+k1 n
x .
k1
(1 x)k =
n0
Solution. When k = 1,
holds.
n+k1
k1
=
n
0
= 1. So (1 x)1 =
Assume that for some positive integer m, (1
Math 239 Winter 2015 Assignment 5 Solutions
1. cfw_5 marks Let cfw_an be the sequence which satises
an 2an1 7an2 4an3 = 0
for n 3 with initial conditions a0 = 3, a1 = 7, a2 = 8. Determine an explicit formula for an .
Solution. The characteristic polynomi
Math 239 Winter 2015 Assignment 1 Solutions
1. cfw_3 marks How many subsets of [20] consists of 2 or 3 odd integers, and any number of even integers?
Solution. There are 10 odd integers in [20], and the number of ways to choose 2 or 3 of them is 10 + 10 .
Math 239 Winter 2015 Assignment 4 Solutions
1. Consider the following set of binary strings.
S = (cfw_1 cfw_0cfw_1 cfw_0) cfw_1
(a) cfw_2 marks List all the binary strings in S of length at most 4.
Solution. , 1, 00, 11, 001, 010, 100, 111, 0000, 0011, 01
Math 239 Fall 2014 Assignment 8 Solutions
1. (a) cfw_2 marks Draw a 3-regular graph that has a bridge.
Solution.
(b) cfw_4 marks Prove that any 3-regular bipartite graph cannot have a bridge.
Solution. Let G be a 3-regular bipartite graph with bipartition
UNIVERSITY OF WATERLOO FINAL EXAMINATION FALL TERM 2003
Surname: First Name: Id.#:
Course Number Course Title
MATH 239 Introduction to Combinatorics Professor Professor Professor Professor Goulden Mosca Schellenberg Verstraete 1:30 2:30 1:30 12:30
Math 239 Winter 2004
1
1. (a) Let k be a given positive integer. Find the generating function for the number of compositions of n having k parts, where each part is an odd positive integer.
(b) [6 marks] Find the generating function for the number
MATH 239 Mid-Term Exam, Nov. 11, 2004
2
This is the midterm examination from Fall 2004. It was both long and challenging. We will try to make our midterm shorter and more routine.
1. (a) [3 marks] Write each of the following rational functions f (
UNIVERSITY OF WATERLOO FINAL EXAMINATION WINTER TERM 2003
Surname: First Name: Id.#: Course Number Course Title MATH 239 Introduction to Combinatorics 01 02 03 Professor Menezes Professor Irving Professor Goulden 8:30 2 10:30 2 1:30 2
Instructor
D
MATH 239 Final Exam, Dec. 20, 2004
2
1. (a) [2 marks] List all compositions of 2, 3 and 4 where each part is a positive integer not equal to 2.
(b) [2 marks] Prove that the generating function for the number of compositions of a positive integer n
UNIVERSITY OF WATERLOO FINAL EXAMINATION FALL TERM 2002
Surname: First Name: Id.#:
Course Number Course Title
MATH 239 Introduction to Combinatorics 01 03 04 05 Goulden Wagner Wagner Schellenberg 2:30 1:30 10:30 9:30 2 2 2 2
Instructor
Date of E