Math 304 Homework 3 comments
1. (a) 96 , everything but 3 is arbitrary; 97 96 (count the complement) or 8 96 (assign
3 in 8 ways everything else in nine ways).
(b) 8 8 7 3 by assigning the value of f(1) in eight ways (it cannot be 5), then
f(2) has eight
Math 304 Homework 7 comments
1. (a) The coefcient of x14 in (x + x2 )10 .
1
.
(1 x)(1
x12 )(1 x24 )(1 x30 )
1
in
.
3 )(1 x5 )(1 x10 )(1 x12 )
(1 x
(b) The coefcient of xk in
(c) The coefcient of x75
(d) The coefcient of x24 in
x6 )(1
x10
.
(1 x)5
(e) Th
Math 304 Homework 8 comments
1. We have that
1
=
(1 x)m+n k
m+n
k
0
xk
and
1
1
=
m (1 x)n
( 1 x)
m
k
k0
k0
k
m
j
=
j= 0
k0
n
k
xk
n
kj
xk
xk .
Since the functions are equal then the coefcients must also be equal and we can
conclude
k
m+n
m
n
=
.
k
j
kj
j=
Math 304 Homework 12 comments
1. This is not possible if all degrees are even. To see this note that if we were able to
remove a single edge and split it into two parts, then each part is itself a graph.
Further the vertex incident to the edge will have o
Math 304 Homework 10 comments
1. There are three types of steps and we will take 12, 9 and 10 steps respectively of
the different types. This comes down to how many ways we can rearrange these
31 steps which can be found via the multinomial coefcient,
31
Math 304 Homework 9 comments
1. (a) Take a derangement of n and consider where n is located. Since it is not
located in its correct position suppose that it is located in the jth slot. Then
we have two cases:
j is located in the nth slot, i.e., the two t
KEY
Math 304 midterm Student name:
This test is closed book and closed notes. No sophisticated calculator is allowed for this test. For full
credit show all of your work (Iegibly!). Each problem is worth 10 points (a total of 60 points).
1. Fill in the
MATH 304, Section A
Name:
Exam 1, Sep 15, 2016
Sec.:
The exam concludes at 10:50pm sharp. You must show all of your work.
Each problem will be graded out of 15 points. The top seven scores will be totaled.
You may use any calculator without wireless capab
Homework 2
MATH 304, Fall 2016
Due Thursday September 08, 2016
You must explain your solutions. Just providing and answer will not be sufficient to get credit for
the problem.
1. (p.62, #19) We are given eight rooks, five of which are red and three of whi
MATH 304, Section A
Name:
Exam 2, Oct 11, 2016
Sec.:
The exam concludes at 10:50pm sharp. You must show all of your work.
Each problem will be graded out of 15 points. The top seven scores will be totaled.
You may use any calculator without wireless capab
Homework 4
MATH 304, Fall 2016
Due Thursday, October 06, 2016
You must explain your solutions. Just providing an answer will not be sufficient
to get credit for the problem.
n
1. Use the proof of Theorem 5.3.3 to show that if A is an antichain of size bn/
Homework 1
MATH 304, Fall 2016
Due Thursday September 01, 2016
You must explain your solutions. Just providing an answer will not be sufficient
to get credit for the problem.
1. (p.20, #1) Show that an m-by-n chessboard has a perfect cover by dominoes if
Homework 3
MATH 304, Fall 2016
Due Thursday September 29, 2016
You must explain your solutions. Just providing an answer will not be sufficient
to get credit for the problem.
1. (p.84, #16) Prove that in a group of n > 1 people, there are two who have the
Math 304 Homework 2 comments
1. For part (a) they should explain why the elements of the map send a multiset to a
set in [k + n 1]. In particular, they should explain why all the elements are unique,
and that the largest an element can be is k + n 1 and t
Math 304 Homework 4 comments
1. Suppose that 1 x1 < x2 < < xk n is such a set. Since each number in this
sequence differs by at least 2 then we have that
1
x1 < x2 1 < x3 2 < < xk (k 1)
n k + 1.
In particular each set we want pairs up with a k element sub
Math 304 Homework 12 comments
1. This is not possible if all degrees are even. To see this note that if we were able to
remove a single edge and split it into two parts, then each part is itself a graph.
Further the vertex incident to the edge will have o
Math 304 Homework 10 comments
1. There are three types of steps and we will take 12, 9 and 10 steps respectively of
the different types. This comes down to how many ways we can rearrange these
31 steps which can be found via the multinomial coefcient,
31
Math 304 Homework 8 comments
1. We have that
1
=
(1 x)m+n k
m+n
k
0
xk
and
1
1
=
m (1 x)n
(1 x)
m
k
k0
k0
k
n
kj
m
j
=
k0
n
k
xk
j=0
xk
xk .
Since the functions are equal then the coefcients must also be equal and we can
conclude
k
m+n
m
n
=
.
k
j
kj
j=0
Math 304 Homework 7 comments
1. (a) The coefcient of x14 in (x + x2 )10 .
1
.
(1 x)(1
x12 )(1 x24 )(1 x30 )
1
.
in
3 )(1 x5 )(1 x10 )(1 x12 )
(1 x
(b) The coefcient of xk in
(c) The coefcient of x75
(d) The coefcient of x24 in
x6 )(1
x10
.
(1 x)5
(e) Th
Math 304 Homework 6 comments
1. (a) By use of inclusion-exclusion the number of such hands is:
4
52
1
13
52 13
4
+
13
2
52 26
4
13
3
52 39
13
= 602, 586, 261, 420.
(b) By use of a generalization of inclusion-exclusion the number of such hands is:
1
1
4
1
Math 304 Homework 5 comments
1. (a) We place the balls in one at a time. For the rst ball we have n choices. Once
we place a ball we gain one new position for the next ball to go, i.e., after
the rst ball it can go above or below the rst ball or in anothe
Math 304 Homework 4 comments
1. Suppose that 1 x1 < x2 < < xk n is such a set. Since each number in this
sequence differs by at least 2 then we have that
1
x1 < x2 1 < x3 2 < < xk (k 1)
n k + 1.
In particular each set we want pairs up with a k element sub
Math 304 Homework 3 comments
1. (a) 96 , everything but 3 is arbitrary; 97 96 (count the complement) or 8 96 (assign
3 in 8 ways everything else in nine ways).
(b) 8 8 7 3 by assigning the value of f(1) in eight ways (it cannot be 5), then
f(2) has eight
Math 304 Homework 2 comments
1. For part (a) they should explain why the elements of the map send a multiset to a
set in [k + n 1]. In particular, they should explain why all the elements are unique,
and that the largest an element can be is k + n 1 and t
Math 304 Homework 9 comments
1. (a) Take a derangement of n and consider where n is located. Since it is not
located in its correct position suppose that it is located in the jth slot. Then
we have two cases:
j is located in the nth slot, i.e., the two t
Math 304 nal
Student name:
This is a take home test. While you may use your book or notes for reference you are not allowed to
use the internet or talk with other people (and students in the class qualify as people). Any attempt to not
follow these rules
Math 304 Homework 5 comments
1. (a) We place the balls in one at a time. For the rst ball we have n choices. Once
we place a ball we gain one new position for the next ball to go, i.e., after
the rst ball it can go above or below the rst ball or in anothe