MAT 344
Quiz 1 Solutions
September 25, 2009
You can see that all of these problems can be solved by extension. But,
what if you had a library of 85 languages, or a casino with thousands of
dice being rolled every hour? You will have to come up with a smar
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MAT344 Quiz 6 TUT0101
1. Use inclusion-exclusion principle to determine how many integers from 1 to 200 are
not divisible by 4 neither by 6.
Let
A4 = cfw_n 200 : 4 divides n,
A6 = cfw_n 200 : 6 divides n.
So |A4 | =
PROBLEM SET 8
Introduction to combinatorics, Fall 2016
Instructor: Diana Ojeda Aristizabal
Problems from the textbook:
Chapter 12: 11, 12, 15, 16
Chapter 7: 3, 7, 10
Additional problems:
How many 4-letter words in the English alphabet begin or end with a
PROBLEM SET 1
Introduction to combinatorics, Fall 2016
Instructor: Diana Ojeda Aristizabal
Problems from the textbook:
Chapter 2: 2, 5, 8, 9, 10, 12
Additional problems:
How many ways are there to arrange the letters in the word BANANA ?
How many ways a
PROBLEM SET 7
Introduction to combinatorics, Fall 2016
Instructor: Diana Ojeda Aristizabal
Problems from the textbook:
Chapter 12: 1, 3, 7, 8.
1. Solution: Use Kruskal. A possible order of edges taken to make a spanning
tree is as follows:
gj, af, kj, ac,
PROBLEM SET 4
Introduction to combinatorics, Fall 2016
Instructor: Diana Ojeda Aristizabal
Problems from the textbook:
Chapter 3: 11, 12, 13, 15, 17, 19
Additional problems:
Suppose that Alice does at least one math problem per day and solves
no more tha
PROBLEM SET 9
Introduction to combinatorics, Fall 2016
Instructor: Diana Ojeda Aristizabal
Problems from the textbook:
Chapter 7: 11, 15, 17
Chapter 8: 1, 4 (a)-(c), 5
Additional problems:
In how many ways can 20 balls be chosen from 12 black balls, 12 w
PROBLEM SET 5
Introduction to combinatorics, Fall 2016
Instructor: Diana Ojeda Aristizabal
Problems from the textbook:
Chapter 5: 1, 2, 3, 5, 6
No additional problems this time.
1
PROBLEM SET 2
Introduction to combinatorics, Fall 2016
Instructor: Diana Ojeda Aristizabal
Problems from the textbook:
Chapter 2: 16, 17, 20, 21, 22, 25, 29, 32
Additional problems:
A manufacturer makes marbles that are identical except for their color,
PROBLEM SET 7
Introduction to combinatorics, Fall 2016
Instructor: Diana Ojeda Aristizabal
Problems from the textbook:
Chapter 12: 1, 3, 7, 8.
Additional problems:
Prove that if G = (V, E) is a simple graph whose maximum vertex degree
is d, then (G) d +
University of Toronto
Faculty of Arts and Science
MAT344 - Introduction to Combinatorics
Term Test
Tuesday, February 28th, 2017 at 10:00 am
Duration: 2 hrs. No aids allowed
Last name .
First name .
Student Number .
- 1. Prove that the graphs G and H in Fi
University of Toronto
MAT344 Midterm Examination
Question 1: Suppose that a computer selects two songs at random, one from each
the album: A Top 10, LtnJzz and B Ltn 41 s most asked. The list of each album is:
A
Song
Musician
B
Song
Musician
1
2
3
4
5
6
7
MAT 344
Quiz 4
November 20, 2009
Question 1: Use the method of generating functions to solve completely
the recursion: for n 1 hn = 2hn1 + 1, and h0 = 1 (10 points).
Note: No credit will be given for a solution that does not use this method.
Answer 1:
hn
MAT 344
Quiz 4 Solutions
November 7, 2009
Question 1: How many ways are there to make an r-arrangement of
pennies, nickels, dimes, and quarters using any number of nickels and
dimes, at least one penny, and an odd number of quarters? (Coins of the
same de
MAT 344
Quiz 2 Solutions
October 19, 2007
Question 1: Prove using a combinatorial argument that
n
2n
2
=
2
2 +n .
2
(10 points)
Some people did not give a combinatorial proof of the identity. You must
learn the combinatorial arguments. Others gave an exam
MAT344 Introduction to Combinatorics
Final Review Part One Solution
Andy Liu
Apr 5, 2017
Note: This note is prepared for the winter course MAT344. There might be
numerous fault arguments/statements/typos. If you spot one, please contact Andy
or you may lo
University of Toronto St. George
MAT344 Spring 2017
Assignment 1 (due on Jan 26)
Problem 1. (10 pts) A graph G is called self-complementary if G is isomorphic to its complement G. Prove that the following property holds: if a graph G with n vertices is se
University of Toronto St. George
MAT344 Spring 2017
Assignment 2 (due on Feb 9)
Note: All necessary figures are on pages 2 and 3. Page 1 contains only problem statements.
Problem 1. (10 pts) Show that the graph G depicted in Figure 1 has an Euler cycle by
University of Toronto St. George
MAT344 Spring 2017
Assignment 4 (due on March 23)
Problem 1. (6 pts) Count the arrangements of the letter in the word CLASSIFICATION
that satisfy a given property (3 separate cases):
a) all possible arrangements, without e
University of Toronto St. George
MAT344 Spring 2017
Assignment 3 (due on March 9)
Problem 1. (5 pts) How many arrangements of the 26 letters of the English alphabet are
there, in which the vowels appear in alphabetic order ? (Here assume Y to be a consona
University of Toronto St. George
MAT344 Spring 2017
Topics for the final: exercises
From the course textbook: Applied Combinatorics by A. Tucker, 6 ed.
Arrangements/Distributions: Section 5.2 (pages 202-203), exercises 40, 41, 42, 49; Section
5.3 (page 21
PROBLEM SET 3
Introduction to combinatorics, Fall 2016
Instructor: Diana Ojeda Aristizabal
Problems from the textbook:
Chapter 2: 27, 33 b,c
Chapter 3: 2, 5, 10
Additional problems:
How many ways are there to distribute 5 apples and 8 oranges to 6 childr
PROBLEM SET 6
Introduction to combinatorics, Fall 2016
Instructor: Diana Ojeda Aristizabal
Problems from the textbook:
Chapter 5: 7, 9, 13, 15, 16, 17, 24, 25, 27.
Additional problems:
Prove that a bipartite graph with an odd number of vertices is not Ha
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QUIZ 5
Introduction to combinatorics, Fall 2016
Instructor: Diana Ojeda Aristizabal
No aids are allowed. You may only use pen, pencil and eraser.
Please have your student ID on the table.
Use the principle of inclusion- ex
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QUIZ 4
Introduction to combinatorics, Fall 2016
Instructor: Diana Ojeda Aristizabal
No aids are allowed. You may only use pen, pencil and eraser.
Please have your student ID on the table.
1. Explain why the graph G below i
Worksheet 5
Introduction to combinatorics, Fall 2016
Instructor: Diana Ojeda Aristizabal
1. a) Consider a connected graph G on n vertices, with (G) = k, where (G) is the minimal
degree of a vertex in G. Prove that between any two vertices there is a path