MAT 226 Discrete Mathematics
Homework Assignment 8 Solutions
1. Solve an 10an 1 25an 2 for n 2 and a0 2 and a1 30 .
Characteristic equation: r 2 10r 25 (r 5)2 0
Roots: r1 r2 5
General solution: an 15n 2n5n
If n = 0, then 2 1 . If n = 1, then 30 51 52 and
MAT 226 Discrete Mathematics
Homework Assignment 9 Solutions
1.
Identify the type of graph (simple graph, multigraph, pseudograph, simple directed
graph, multiple directed graph) in each of the following.
a) Section 10.1, Problem 4
Multigraph
b)
Section 1
MAT 226 Discrete Mathematics
Homework Assignment 11 Solutions
1.
Section 10.5, Problem 19, but add a second edge from d to b (in that order). Only
determine whether there is an Euler circuit (justify your answer), and, if so, show
one.
There is no Euler c
MAT 226 Discrete Mathematics
Homework Assignment 12 Solutions
1.
Section 10.7, Problem 4
a
c
b
d
e
2.
Section 10.7, Problem 5
No. This graph is K3.3.
a
c
e
b
d
a
e
4.
f
b
3.
d
c
f
Section 10.7, Problem 6
Section 10.7, Problem 12
Eulers formula says that v
Solving Linear Recurrence Relations
Linear homogeneous recurrence relation of degree k with constant coefficients:
A recurrence relation of the form an c1an 1 c2an 2 . ck an k , where c1, c2 ,., ck are
real numbers and ck 0 .
The characteristic equation o
MAT 226 Discrete Mathematics
Homework Assignment 13 Solutions
1.
Section 11.1, Problem 3
a) a
b) a, b, c, d, f, h, j, q, t
c) e, g, i, k, l, m, n, o, p, r, s, u
d) q, r
e) c
f) p
g) f, b, a
h) e, f, l, m, n
2.
Section 11.1, Problem 16
3.
Section 11.1, Pro
Theorem
Wednesday, February 04, 2015
10:58 AM
The number of permutations of n objects
n1 of type 1 (indistinguishable)
n2 of type 2 (indistinguishable)
.
.
.
nk of type k (indistinguishable)
Notes Page 1
Distinguishable objects among labeled boxes
Wednesd
Notion of Set
Wednesday, January 14, 2015
10:23 AM
Notion of a Set: A set is an unordered collection of
distinct objects
Ex: A = cfw_1, c, 13
a set B is a subset of Set A if every object in B is also an
object in A
B <= A, symbols
Ex: B = cfw_2,6,12
A = s
Choosing
Monday, January 26, 2015
10:57 AM
Ex: In how many ways can we choose 5 students from a
group of 23 to form a committee?
C(23,5)
Notation: Binomial Coefficient
Ex: There is a group of 8 republicans and 12 democrats
1)
2)
3)
4)
5)
In how many ways
MAT 226 Discrete Mathematics
Homework Assignment 7 Solutions
1. Find a recurrence relation for the number of permutations of a set with n elements.
(Dont forget to include a statement about the valid values of n, as well as the initial
conditions.)
Let an
MAT 226 Discrete Mathematics
Homework Assignment 4 Solutions
1. How many nonnegative integer solutions are there for the equation
x1 x2 x3 x4 x5 x6 32 :
a)
Total (without additional restriction)?
37
435,897
5
b)
Such that x2 3, x4 5, x5 2 ?
27
80,73
Extra Credit: Use Dijkstra's Algorithm 1.
(a) Find the length of the shortest path from a to z. (b) Find the shortest path from a to z.
2.
(a) Find the length of the shortest path from a to z. (b) Find the shortest path from a to z. (c) Find the length of
MAT 226 Exam 2 Practice Problems
These are just supplemental problems for you to work on. When preparing for the test, also look over your
homework problems, quizzes, and notes.
1. Prove that 12 + 32 + 52 + + (2n + 1)2 =
(n + 1)(2n + 1)(2n + 3)
for all n
MAT 226 Discrete Mathematics
Homework Assignment 2 Solutions
1. Show that in any set of 29 distinct positive integers, there are at least three with the
same remainder when divided by 13.
Group the numbers according to the remainders upon division by 13.
MAT 226 Discrete Mathematics
Homework Assignment 3 Solutions
1. A 5-card poker hand is dealt from a standard 52-card deck. Calculate the probability
that the hand is:
(a) A flush (all five cards from the same suit).
13
There are four ways to pick a suit
MAT 226 Discrete Mathematics
Exam #3 Solutions
This is a take-home exam. It is not a group exam and you must complete the exam on
your own. You may use your books and notes, but no others resources whether it be
human or electronic (i.e., the Internet).
1
MAT 226 Discrete Mathematics
Exam #1 Solutions
Show any necessary supporting work. Express conclusions clearly, using correct
language and notation. Leave answers in symbolic form except for Problem 1.
(2 pts.)
1. Evaluate both of the following: P(8, 3) a
50 l U‘tl mqj
MAT 226 Discrete Mathematics 510?“
Exam #2
Each of the problems are worth four points.
1.
a) Give an example of a linear ho
mogeneous recurrence relation with constant
coefficients of degree 3.
coefficients of degree 2.
an a201,".I + 3a“
1.
MAT 226 Discrete Mathematics
Homework Assignment 1 Solutions
1. Textbook: 6.1, Problem 12
There are 2n bit strings of length n, so the number of bit strings of length 6 or less (not
counting the empty string) is: 21 + 22 + 23 + 24 + 25 + 26 = 126
2. Textb
MAT 226 Discrete Mathematics
Homework Assignment 5 Solutions
1. Find a formula for
1 1
1
. n and justify your formula with a proof.
2 4
2
First note:
1
1
1 1
2
2
1 1 3
1
1 2
2 4 4
2
1 1 1 7
1
1 3
2 4 8 8
2
So a conjecture might be that
1 1
1
1
. n 1
11a. Exam
Tuesday, February 10, 2015
6:56 PM
Notes Page 1
Nicholas Finch
Friday, February 13, 2015
10:07 AM
Let n be the first
element
To show that the equation
Holds true
Based on the first element
The evaluation holds true
Let k be a positive intege