Assessment 2 Top of Form Bottom of Form Content · Print · Counting, Combinations, and Permutations · · Details · Attempt 1Available · Attempt 2NotAvailable · Attempt 3NotAvailable · Toggle Drawer Overview Refresh a company's computer network memory with respect to number representation conversions, decimal to binary and hexadecimal (and vice versa), using your ability to apply number representation and theory. Then, use discrete probability to assess the risk of a hacker foiling the company network's RSA encryption. The assessment focuses on number theory, discrete probability theory, counting rules, permutations, and combinations. Show Less By successfully completing this assessment, you will demonstrate your proficiency in the following course competencies and assessment criteria: · Competency 2: Apply the methodologies of discrete math. · Convert numbers to different representations. · Compute combinations and/or permutation. · Compute discrete probability. · Combine discrete probability. · Competency 4: Apply discrete math methods and tools to solve problems encountered in a work setting. · Apply number theory to encryption. Competency Map Check Your ProgressUse this online tool to track your performance and progress through your course. · Toggle Drawer Context The Assessment 2 Context document contains additional information about set and probability theory, permutations and combinations, and cryptography. · Toggle Drawer Resources Suggested Resources The following optional resources are provided to support you in completing the assessment or to provide a helpful context. For additional resources, refer to the Research Resources and Supplemental Resources in the left navigation menu of your courseroom. Capella Resources Click the links provided to view the following resources: · Assessment 2 Context. Show Less Capella Multimedia Click the links provided below to view the following multimedia pieces: · The Counting Principle | Transcript. · This presentation introduces the following topics: · Multiplication and addition principles. · Permutations and combinations. · Catalan numbers. · Discrete probability. · Conditional probability. · Pigeonhole principle. Library Resources The following e-book from the Capella University Library is linked directly in this course: · Koshy, T. (2004). Discrete mathematics with applications. Burlington, MA: Elsevier Academic Press. · Chapter 6. Course Library Guide A Capella University library guide has been created specifically for your use in this course. You are encouraged to refer to the resources in the MAT-FP2051 – Discrete Mathematics Library Guide to help direct your research. Bookstore Resources The resource listed below is relevant to the topics and assessments in this course and is not required. Unless noted otherwise, this resource is available for purchase from the Capella University Bookstore. When searching the bookstore, be sure to look for the Course ID with the specific –FP (Fl ...

1. Assessment 2
Top of Form
Bottom of Form
Content
· Print
· Counting, Combinations, and Permutations
·
· Details
· Attempt 1Available
· Attempt 2NotAvailable
· Attempt 3NotAvailable
· Toggle Drawer
Overview
Refresh a company's computer network memory with respect to
number representation conversions, decimal to binary and
hexadecimal (and vice versa), using your ability to apply
number representation and theory. Then, use discrete
probability to assess the risk of a hacker foiling the company
network's RSA encryption.
The assessment focuses on number theory, discrete probability
theory, counting rules, permutations, and combinations.
Show Less
By successfully completing this assessment, you will
demonstrate your proficiency in the following course
competencies and assessment criteria:
· Competency 2: Apply the methodologies of discrete math.
· Convert numbers to different representations.
· Compute combinations and/or permutation.
· Compute discrete probability.
· Combine discrete probability.
· Competency 4: Apply discrete math methods and tools to solve
problems encountered in a work setting.
· Apply number theory to encryption.
Competency Map

2. Check Your ProgressUse this online tool to track your
performance and progress through your course.
· Toggle Drawer
Context
The Assessment 2 Context document contains additional
information about set and probability theory, permutations and
combinations, and cryptography.
· Toggle Drawer
Resources
Suggested Resources
The following optional resources are provided to support you in
completing the assessment or to provide a helpful context. For
additional resources, refer to the Research Resources and
Supplemental Resources in the left navigation menu of your
courseroom.
Capella Resources
Click the links provided to view the following resources:
· Assessment 2 Context.
Show Less
Capella Multimedia
Click the links provided below to view the following
multimedia pieces:
· The Counting Principle | Transcript.
· This presentation introduces the following topics:
· Multiplication and addition principles.
· Permutations and combinations.
· Catalan numbers.
· Discrete probability.
· Conditional probability.
· Pigeonhole principle.
Library Resources
The following e-book from the Capella University Library is
linked directly in this course:
· Koshy, T. (2004). Discrete mathematics with applications.
Burlington, MA: Elsevier Academic Press.
· Chapter 6.

3. Course Library Guide
A Capella University library guide has been created specifically
for your use in this course. You are encouraged to refer to the
resources in the MAT-FP2051 – Discrete Mathematics Library
Guide to help direct your research.
Bookstore Resources
The resource listed below is relevant to the topics and
assessments in this course and is not required. Unless noted
otherwise, this resource is available for purchase from the
Capella University Bookstore. When searching the bookstore,
be sure to look for the Course ID with the specific –FP
(FlexPath) course designation.
· Johnsonbaugh, R. (2018). Discrete mathematics (8th ed.). New
York, NY: Pearson.
· Chapter 6, "Counting Methods and the Pigeonhole Principle,"
sections 6.1, 6.2, 6.3, 6.5, and 6.6, are particularly useful for
your work in this assessment. Topics in these sections include
permutations, combinations, discrete probabilities, and discrete
probability theory.
· Assessment Instructions
Assume you help to oversee your company's computer network.
As such, it is important for you to understand and be able to
apply number representation and number theory, as well as
other IT concepts.
Part 1: Number Representation (application to binary encoding)
and Combinatorics (application to IP network addressing)
Note: For each of the following, you must show your work for
credit.
Given your responsibilities, you decide to refresh your memory
with respect to number representation conversions: decimal to
binary and hexadecimal (and vice versa). In the following
questions, the base is denoted as a subscript. For example, 1510
is 15 in decimal (base 10), 00112 is 3 in binary (base 2), and
1A16 is 26 in hexadecimal (base 16).
7. What is the decimal representation of 100011012 ?
7. What is the decimal representation of FFC616 ?

4. 7. What is the binary representation of 17C616 ?
7. What is the hexadecimal representation of 111110002 ?
According to the IP internet protocol, each IP address is
represented as a binary string. In IPv4 (Internet protocol version
4), a 32-bit binary string is used. For example,
00000011.00000111.00000000.11111111, which is often
presented in dotted decimal: 3.7.0.255.
7. In mathematics, the study of combinations refers to the
number of ways one can select items from a group disregarding
order; the study of permutations refers to the number of ways
one can permute, or arrange, items into a sequence. Given that
each entry in a binary string must be either a 1 or a 0, what is
the total number of addresses that can be encoded using a 32-bit
binary string? Is this a combination or permutation problem?
Justify your answer.
7. In IPv6, 128 bit, binary strings are used for addressing. How
many addresses can be encoded using 128 bits? Is this a
combination or permutation problem? Justify your answer.
7. In IPv4, how many addresses contain exactly eight 1s?
Part 2: Number Theory and Discrete Probability (application to
encryption)
Note: For each of the following, you must show your work for
credit. Some questions also require you to justify your answer.
Network security and encryption is also a concern of a network
administrator. Many encryption schemes are based on number
theory and prime numbers; for example, RSA encryption. These
methods rely on the difficulty of computing and testing large
prime numbers. (A prime number is a number that has no
divisor except for itself and 1.)
For example, in RSA encryption, one must choose two prime
numbers, p and q; these numbers are private but their product, z
= pq, is public. For this scheme to work, it is important that one
cannot easily find p or q given z, which is why p and q are
generally large numbers.
1. Choose an example of p and q and compute their product z.
Justify your selection.

5. 2. Assume that you wish to make a risk assessment and you
wish to determine how probable it may be for a hacker to
determine p and q from z. You wish to use discrete probability
for this computation. For the sake of example, you choose to
assess z = 502,560,410,469,881. Say that a hacker will attempt
to find p and thus q by randomly selecting a potential divisor
and testing to see if it divides 502,560,410,469,881. (You know
that p = 15,485,867 and q = 32,452,843, but the hacker does
not.) For example, the hacker may choose 1021; however, upon
inspection the hacker will see that 1021 does not divide z.
For all questions below, please show all your work and/or
justify your answers.
a. Given this problem, what is the sample space of the problem?
Hint: In this context, the sample space is the set of all possible
values that the hacker may select.
b. Given the sample space defined above, what events
correspond to a successful guess by the hacker? Hint: An event
is a subset of the sample space.
c. Given the above, what is the probability that the hacker will
successfully guess p and/or q?
d. Assume the hacker selects five numbers to test.
i. What is the probability that all five attempts will fail? Show
your work.
ii. What is the probability that one of the five attempts will
succeed? Show your work.
Assessment 2 Context
First, let us think about basic discrete (non-continuous)
probability using a standard deck of playing cards. We know
there are 52 cards, 13 different values from 2 to ace, and four
suits (that is, clubs, spades, hearts, and diamonds). Many
probability questions can be asked about a deck of cards. For

6. example, what is the probability that a card you choose is a
queen?
P(card = queen).
We know there are 52 cards and four queens.
P(card = queen) = 4/52.
We can extend this concept to multiple events. What is the
probability of choosing two cards and both are queens?
P(card1 = queen) AND P(card2 = queen).
Notice the AND. A concept called the multiplication principle
that considers the results of more than one event occurring
together.
P(card1 = queen) AND P(card2 = queen) = 4/52 * 3/51 =
12/2,652.
We used multiplication in this case and did not replace the first
card after choosing it.
Similarly, the addition principle is used for independent events
such as the probability of choosing a card and it being a queen
OR a king.
P(card1 = queen) OR P(card1 = king) = 4/52 + 4/52 = 8/52.Set
Theory and Probability Theory
Set theory and probability theory are interrelated. An event can
be thought of as a set of possible outcomes. For example,
rolling one six-sided die is an event. This event is discrete
because each outcome is a whole number (that is, no fractions
or decimals).
The sample space S = {1, 2, 3, 4, 5, 6}.
Rolling a die is an event that offers a set of six possible
outcomes as noted above. This set can be named E. A sample
space S, also called the universe of values, is the set of all
possible outcomes. Given both E and S, the probability of event
E occurring can be computed as follows:
P(E) = |E| / |S|
An event is the outcome or outcomes of a trail. For any event E
in a given sample space, a function P, known as the probability
function, assigns a value to P( E).
Note that: 0 ≤ P( E) ≤ 1.

7. This tells us that the probability of any event must be between 0
and 1 (or 0% chance to 100% chance).
For example, let S be a sample space with all possible values of
rolling a six-sided die:
S = {1, 2, 3, 4, 5, 6}.
Let E = {4}.
P(E) = |{1}| / |{1,2,3,4,5,6}| = 1/6, which is between 0 and
1.Permutations and Combinations
A permutation is the number of possible ways of rearranging a
discrete set (no decimals or fractions) of objects. For example,
if someone gave you four pictures to hang on the wall and asked
how many ways you could hang them in a straight line, you
would say:
P(4, 4) = 4! = 4 * 3 * 2 * 1 = 24 ways.
The P(4, 4) notation reads, "4 items, permute all 4 of them."
The 4! reads, "4 factorial" and is a discrete function. The
general notation of permutation is P( n, r), given n objects, how
many ways can you permute r of them.
P(n,r) = n!/(n-r)!
A combination is the number of ways you can select a set of
objects when the order does NOT matter. The notation is C( n,
k) and is read "n choose k".
C(n,k) = n!/[(n- k)! k!]Cryptography
The word cryptography comes from the word cryptic, which
means hidden. Cryptography focuses on encrypted and
decrypted information that is intended only for the receiver. E-
mail systems, such as Hotmail, claim to use encryption.
Cryptography uses many areas of discrete math, including
modulus, prime factorization, and greatest common divisors.