21-228 Discrete Mathematics
Assignment 7
Due Wed Apr 13, at start of class
Notes: Collaboration is permitted until the writing stage. Please justify every numerical answer with an
explanation. The base value of each question is 10 points.
1. Suppose that
21-228 Discrete Mathematics
Test 3: Solutions
1. Let H be a 3-uniform hypergraph with with m edges. Prove that there is a partition of the vertices
into three disjoint sets V1 V2 V3 such that the number of crossing hyperedges is at least 2 m. (In
9
the se
21-228 Discrete Mathematics
Assignment # 3
Solutions
1. 6 519 .
We show that the number of strings of length k in which no two consecutive entries are
the same is 6 5k1 for all positive integers k by induction on k.
Of course, there are six 1-element stri
21-228 Discrete Mathematics
Assignment # 6
Due: Friday, March 26
1. A bowl contains n cherries, exactly m of which have had their stones removed. A
pig eats p cherries chosen at random without announcing how many contain stones.
Subsequently, a cherry is
21-228 Discrete Mathematics
Assignment # 8
Due: Friday, April 9
1. Let G be a connected r-regular graph of order n such that G is also connected.
(a) Show that either G or G is Eulerian.
(b) Show that either G or G is Hamiltonian
2. A connected graph G =
21-228 Discrete Mathematics
Assignment # 3
Due: Friday, February 5
1. Find the number of strings of length 20 in which each entry is from the set cfw_1, 2, 3, 4, 5, 6
and no two consecutive entries are the same.
2. Give a combinatorial proof of the follow
21-228 Discrete Mathematics
Assignment # 4
Due: Friday, February 19
1. A class of n children take off their shoes. The shoes are then distributed to the children
so that each child gets one left shoe and one right shoe. Give a formula for the number
of wa
21-228 Discrete Mathematics
Assignment # 2: Solutions
1. We have
! "
n
n(n 1)(n 2) (n k + 1)
=
k(k 1)(k 2) 1
k
!
"
#n$ !n 1" !n 2"
nk+1
=
.
k
k1
k2
1
However, for i = 1, . . . , k 1 it is easy to check that we have
It follows that
n
ni
.
k
ki
!
"!
"
!
" !
21-228 Discrete Mathematics
Assignment # 2
Due: Friday, January 29
1. Let n, k be positive integers such that 1 k n 1. Prove the following inequalities:
! "
nk
n
nk
.
k
kk
k!
2. There are 4 candidates in an election. A population of m people cast ballots
21-228 Discrete Mathematics
Assignment 8
Due Fri Apr 29, at start of class
Notes: Collaboration is permitted until the writing stage. Please justify every numerical answer with an
explanation. The base value of each question is 10 points.
1. Construct a r
21-228 Discrete Mathematics
Assignment 6
Due Fri Apr 1, at start of class
Notes: Collaboration is permitted until the writing stage. Please justify every numerical answer with an
explanation. The base value of each question is 10 points.
1. We say that a
21-228 Discrete Mathematics
Assignment 5
Due Fri Mar 18, at start of class
Notes: Collaboration is permitted until the writing stage. Please justify every numerical answer with an
explanation. The base value of each question is 10 points.
1. Solve the rec
21-228 Discrete Mathematics
Assignment 4
Due Fri Feb 25, at start of class
Notes: Collaboration is permitted until the writing stage. Please justify every numerical answer with an
explanation. The base value of each question is 10 points.
1. Prove that if
21-228 Discrete Mathematics
Assignment # 6: Solutions
1. We can express this experiment as a probability space as follows. Let X = cfw_1, . . . , n.
This denotes the cherries. We will assume that cherries 1, . .!. , m
" do not contain stones.
Let be the s
21-228 Discrete Mathematics
Assignment # 4 : Solutions
1. Let Sn be the set of permutations of the set cfw_1, 2, . . . , n. Let be the set of all ordered
pairs (, ) such that both , Sn . We think of as the rearrangement of the left
shoes and as the rearra
21-228 Discrete Mathematics
Assignment # 7: Solutions
1. Our probability space is the collection of all Red-Blue colorings of the edges of Kn where
n is arbitrary. We color each edge independently at random so that
P r(e is Red) = p
for all e
![n]"
2
and
21-228 Discrete Mathematics
Test 2: Solutions
1. Consider the Fibonacci recursion:
F0 = 0 ;
Prove or disprove:
F1 = 1 ;
n
Fn = e 2 1 ,
Fn = Fn1 + Fn2 .
for all positive integers n.
Here, x represents x rounded up.
Solution. The answer is no. Recall from c
21-228 Discrete Mathematics
Po-Shen Loh
April 2011
Test 3
Problem
Score
1
2
3
4
Total
Name:
This 50-minute exam is open-book and open-notes. Calculators are permitted. Please write your
answers in the space provided, and indicate clearly if you use the ba
21-228 Discrete Mathematics
Po-Shen Loh
7 February 2010
Test 1
Problem
Score
1
2
3
4
Total
Name:
This 50-minute exam is open-book and open-notes. Please write your answers in the space provided, and
indicate clearly if you use the back of a page for addit
21-228 Discrete Mathematics
Po-Shen Loh
25 March 2011
Test 2
Problem
Score
1
2
3
4
Total
Name:
This 50-minute exam is open-book and open-notes. Calculators are permitted. Please write your
answers in the space provided, and indicate clearly if you use the
21-228 Discrete Mathematics
Test 1: Solutions
1. Determine the limit
lim
n
4n
n
4n
2n
.
Solution. Recall Stirlings approximation:
t!
t
e
t
2t,
i.e.,
lim
t
t t
e
t!
= 1.
2t
We rst simplify the binomial coecients:
4n
n
4n
2n
=
(4n)!
n!(3n)!
(4n)!
(2n)!(2n)
21-228 Discrete Mathematics
Course Review 3
This document contains a list of the important definitions and theorems that have been
covered thus far in the course. It is not a complete listing of what has happened in lecture.
The sections from the book tha
21-228 Discrete Mathematics
Course Review 2
This document contains a list of the important definitions and theorems that have been
covered thus far in the course. It is not a complete listing of what has happened in lecture.
The sections from the book tha
21-228 Combinatorics
Course Review 4
This document contains a list of the important definitions and theorems that have been
covered thus far in the course. It is not a complete listing of what has happened in lecture.
The sections from the book that corre
21-228 Discrete Mathematics
Course Review 1
This document contains a list of the important definitions and theorems that have been
covered thus far in the course. It is not a complete listing of what has happened in lecture.
The sections from the book tha
21-228 Discrete Mathematics
Assignment # 8: Solutions
1. (a) We begin with an observation. If G is an r-regular graph of order n then the sum
of the degrees, which is twice the number of edges and therefore even, is equal to
rn. So either r is even or n i
21-228 Discrete Mathematics
Assignment # 7
Due: Friday, April 2
1. Prove that for all integers n and p [0, 1] we have
! "
! "
!
n (k2)
n
R(k, l) > n
p
(1 p)(2) .
k
!
Hint: Consider a random coloring of edge set of Kn in which we color each edge independ
21-228 Discrete Mathematics
Assignment 3: Solutions
1. A set S of numbers is called a Sidon set if it has the property that for every distinct a, b, c, d S,
the sums a + b and c + d are dierent. For example, cfw_1, 2, 4, 8 is Sidon, but cfw_1, 2, 3, 4 is
21-228 Discrete Mathematics
Assignment 2: Solutions
1. How many rearrangements of the word DOCUMENT have the three vowels all next to each other?
For example, DOEUCMNT counts, but not DOCUEMNT.
Solution. Consider the three vowels as a single block. Then t
5 D
5 D 5D ( `
x'!
's 6 6 9
Dq# tD % r % 1 Q % D % 2 HG 4#
I A %21 )
Q Y r Q
s Y Q Y | Q Y Q Y Q Y
T QR T QR T QR z T QR w T QR Q
QR
A ( Q# Q "vvT (Q# t %
's
T ' ws 6
' r s r 6 1
1
p 9 1 1 A17
T v4# yD % 2 n9 41 1 n6 oov"v1 h ( % p 1 p t2
!'
.1 1 Ha)7=olm'u.uuc.n .
I [I\=h
" i /
A )(b
deate (JIM e 618'th in, B , . ,
In W0 introduce, an fwd/(0107 random vahablx.
'f VOLEA cfw_(amb;
use , 0
a
f "0-1 3 MI, 18"? 5 denote NATL? +ns Maid (my-Eb?
. Sf 11 ' d ' " ;
4 = 42%;: (L; W W at m w a,