Math 461
Solutions for Homework #5
November 5, 2012
1
1. (This is #35 on p. 107) Let an be the number of compositions of n into parts that are larger
than 1. Express an by an1 and an2 .
Solution 1: We claim that an = an1 + an2 . Indeed, each such composit
Math 461
Solutions for Homework #7
1
November 30, 2012
1. Let T be a tree on n vertices such that the degree of each vertex is either 1 or 3.
(a) Show that n is even.
Solution: Since the number of odd-degree vertices in any graph (in particular, in a
tree
Math 461
Homework #3
due Monday, October 13
1
Reading assignment:
After Mondays lecture, read carefully Lecture 6 notes and 3.2 in the Invitation to Discrete
Math book.
After Wednesdays and Fridays lectures, read carefully Lecture 7 and Lecture 8 notes
Math 461
Homework #6
due Friday, November 9
1
Reading assignment: Read Theorem 9.2 and Corollary 9.3 in Section 9.1; start reading Sections 9.2 and 9.4.
Written assignments:
1. Which complete graphs Kn have Eulerian trails?
2. In a certain country, 100 ro
Math 461
Solutions for Homework #6
November 9, 2012
1
1. Which complete graphs Kn have Eulerian trails?
Solution: Observe that in Kn all vertices have degrees n 1. Observe also that n 1 is
even if and only if n is odd. Now, according to the Euler theorem,
Math
October L5, 2074
First Midterm
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Math 461A
Partial Solutions to review problems October 13
1
1. Show that in every graph with n 2 vertices, there are two vertices that have the
same degree.
Solution: Let G be a graph with n 2 vertices. Then the degree of each vertex of G
is an integer be
Math 461
Homework #4
due Monday, October 29
1
Reading assignment: Read carefully Chapter 7 (you may skip Theorem 7.6) and glance through
pages 95-96 of Section 5.2.
Written assignments:
1. (this is #24 on p. 144) How many three digit positive integers are
Math 461
Homework #2
due Monday, October 6
1
Reading assignment:
After Mondays lecture, read carefully Lecture 3 notes. You may also want to take a look at 2.4
in the book by L. Lovsz, J. Pelikn, and K. Vesztergombi.
a
a
After Wednesdays lecture, read c
Math 461
Solutions for Homework #4
October 29, 2012
1
1. (this is #24 on p. 144) How many three digit positive integers are divisible by at least one of
six and seven?
Solution: Let
S = cfw_99 < x 999 : x Z,
S6 = cfw_x S : 6|x,
and S7 = cfw_x S : 7|x.
We
Math 461
Solutions for Homework #1
October 1, 2012
1
1. Each box in a 3 3 arrangement of boxes is lled with one of the numbers 1, 0, 1. Prove
that of the eight possible sums along the rows, the columns, and the diagonals, two sums
must be equal.
Solution
Math 461
Solutions for Homework #3
October 14, 2014
1
1. How many four-digit positive integers are there in which all digits are dierent? (We do not allow the
rst digit to be zero.)
As we do not allow the rst digit to be zero, there are 9 choices for the
Math 461
Solutions for Homework #2
October 7, 2014
1
1. Let a0 = 0, a1 = 1, and let an+2 = 6an+1 9an for n 0. Prove that an = n 3n1 for all n 0.
The proof is by (strong) induction on n. If n = 0 or n = 1 the statement does hold: 0 = 0 31 and
1 = 1 30 . No
Math 461
Homework #4
due Monday, October 27
1
Reading assignment:
After Fridays lecture, read carefully Lecture 11 notes and glance at p. 69 in the textbook.
After Mondays lecture, read carefully Lecture 12 notes and p. 69 in the book; then skim through
Math 461
Homework #1
due Monday, September 29
1
Homework assignments in this class will be usually due on Monday. The importance of doing
all the homework cannot be overemphasized. Mathematics cannot be learned passively. Even professional mathematicians
Math 461
Homework #3
Due October 22, 2012
1
Reading assignment: Read Sections 5.1 and 7.1.
Written assignments:
1. p. 81, #(50) If we expand the expression (x1 + x2 + x3 + x4 )6 , what will be the largest
coecient that occurs?
2. p. 107, #24: Find the num
Math 461
Homework #1
due Monday, October 1
1
Reading assignment: Read the rst two chapters of the book. I expect to cover Chapter 1 on
Mon-Wed, Sep 2426, and Chapter 2 on Fr-Mon, Sep 2628.
Written assignments:
1. Each box in a 3 3 arrangement of boxes is
Math 461
Solutions for Homework #2
October 8, 2012
1
1. p. 53, #(30) How many four-digit positive integers are there in which all digits are dierent?
Solution: As we do not allow the rst digit to be zero, there are 9 ways to choose the rst
digit. Once the
Math 461
Solutions for Homework #3
October 22, 2012
1
1. p. 81, #(50) If we expand the expression (x1 + x2 + x3 + x4 )6 , what will be the largest
coecient that occurs?
6!
Solution: All coecients in the expansion are of the form a!b!c!d! , where a, b, c,
Math 461
Homework #5
due Monday, November 5
1
Reading assignment: Read pages 95-96 of Section 5.2, and Section 9.1: skip Theorem 9.2 and
Corollary 9.3 for now, but make sure you read everything else in this section; also make sure you
know how to solve Pr
Math 461
Homework #7
due Friday, November 30
1
Reading assignment: Read 10.1, the rst two paragraphs of 10.2, and the solution to Problem
#(5) on p. 234 (the solution itself is on pp. 239-240).
Written assignments:
1. Let T be a tree on n vertices such th
Math 461A
First midterm Solutions
October 17, 2012
1
General comments: Please review basic high-school algebra. For instance, make sure you
know the order of operations, and understand the dierence between (i) (2n)! and 2 n!, (ii)
2
n
2n and 22 2n , (iv)