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 bas
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 chec
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 candidate
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 bas
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
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 bas
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 bas
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 e
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.
Soluti
21-228 Discrete Mathematics
Assignment 1: Solutions
1. A class has 26 students, named Alice, Bob, Charlie, . . . , Zed. How many dierent Facebook-friendship
networks are possible? For example, one pos
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 bas
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
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 bas
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
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 ar
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 the
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
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
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 a
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 cher
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
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
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
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 hap
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
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 announc
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
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 bas
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 bas
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