MATH125: Unit 8 Submission Assignment Answer Form: Counting Techniques

This document is a comprehensive solution for the MATH125 Unit 8 submission assignment, focusing on counting techniques. The assignment covers key concepts such as permutations and combinations, including their formulas and differentiating factors. It delves into real-world applications, like calculating the number of possible license plates based on specific state formats and character rules. Furthermore, the assignment explores probability and odds, requiring the differentiation between them, calculating ratios, and applying them to scenarios involving dice rolls. It includes theoretical, empirical, and experimental probability calculations, along with likelihood scale analysis. The solution provides step-by-step calculations and explanations for each problem, demonstrating a thorough understanding of the concepts.

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MATH125: Unit 8 Submission Assignment Answer Form
Counting Techniques
ALL questions below must be answered. Show ALL step-by-step calculation. Upload this
modified Answer Form to the intellipath Unit 8 Submission lesson. Make sure you submit your
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Part A: Combinations and Permutations
1. Differentiate between permutations and combinations. How are they different? What is
the formula for each? (15 points total for Question 1)
How are they
different?
(5 points)
A permutation is an arrangement of
items that doesn’t not take into account
the order of items in the list, on the
other hand combination is an
arrangement of items taking into
account the order in which the items
appear on the list
Permutation Formula
(5 points)
Permutation of n objects that are
selected r at a time is given by the
formula P ( n , r ) = n !
( n−r ) !
Combination Formula
(5 points)
Combination of n objects that are
taken r at a time are given by:

C ( n , r ) = n!
( n−r ) !r !
2. Each state has a standard format for license plates that includes a set number of
alphanumeric characters. For this assignment, you can insert a picture of your state’s non-
personalized license plate or provide a sample of the format in text. (20 points total for
Question 2)
Your State’s Name
(1 point)
Alaska
Picture of a License
Plate from Your
State
(or a Sample)
(1 point)
Describe the
Character Rule for
Your State’s Non-
The non -personalized plates ought to
have 3 letters followed by 3 numeric
values.

Personalized License
Plates
(1 point)
a. Determine the number of different license plates that can be created using the
following format. Assume that a license plate consists of seven alphanumeric
characters using numbers (0–9) and capital letters (A–Z). Find how many unique
license plates can be printed using any of the 36 alphanumeric characters without
duplication in each of the seven alphanumeric characters, i.e., no alphanumeric
character appears more than once in any license plate. (This question is NOT
related to your state’s license plates in the initial 3 parts above).
Is this a permutation
or combination?
Why?
(2 points)
This is permutation since the order in
which the alphanumeric characters are
arranged does not matter
What formula from
Question 1 will you
use to solve the
problem?
(1 point)
P ( n , r )= n !
( n−r ) !
Number of unique
license plates that can
be printed:
(2 points)
P ( 36 ,7 ) = 36 !
( 36−7 ) !
¿ 4.20723072∗1010

Show your work here (2 points):
b. You and a friend are witnesses of a car accident in your state. But you can only
remember a few of the first alphanumeric characters on the license plate.
How many
alphanumeric
characters do you
remember?
(1 point)
4
What are the
characters at the
beginning?
(1 point)
FWD 3
How many license
plates start with these
alphanumeric
characters?
(2 points)
100
Show your work here (2 points):

The license plate from Alaska state are arranged in 3 letters followed by 3 numbers.
Having identifies the 3 letters and one of the numbers the other two can be combined in
10∗10=100 ways
This means 100 licensees can start with the identified numeric
How many license
plates have been
eliminated?
(2 points)
175759
Show your work (2 points):
The 1st 4 letters can be arranged in the following ways
26∗26∗26∗10=175760
Since we have identified 1 order the rest of 175759 have been eliminated
3. Your community has asked you to help the YMCA sports director organize a season of
sports. You need to put together the teams. For the soccer teams, athletes signed up for
one of three different age groups (Little Tykes, Big Kids, and Teens). (15 points total for
Question 3)
Is this a permutation
or combination?
Why?
(2 points)
This is permutation. The students
signed in each group can be selected
in any order to form a team.
What formula from
Question 1 will you
use to solve the
problem?
(1 point)
P ( n , r )= n !
( n−r ) !

Use the formula and values, from question above, to answer the following:
How many kids
signed up for “Little
Tykes”?
(1 point)
60
How many kids
signed up for “Big
Kids”?
(1 point)
50
How many kids
signed up for Teens?
(1 point)
70
How many total
students signed up for
soccer?
(1 point)
60+50+70=180
How many different
ways can you create
teams of 10 for the
“Little Tykes” grade
level?
(2 points)
2.735898472∗1017

Show your work here: (2 points)
Total number of children that signed up for the group was 60
Selecting a team of 10 from 60
P ( 60 ,10 ) = 60 !
( 60−10 ) ! =2.735898472∗1017
Show your work here: (2 points)
The total number of people who signed for soccer was 180
Now selecting a team of 10from this number
P ( 180 ,10 )= 180 !
(180−10 ) ! =180∗179∗178∗177∗176∗175∗174∗173∗172∗171=2.768148789∗1022
Part B: Probabilities and Odds
4. For this set of exercises, you will need a single standard six-sided die. If you do not have
one, you can use a virtual die: https://www.random.org/dice/ (40 points total for
Question 4)
a. First, differentiate between odds and probability.
If age levels did not
matter, how many
different ways can
you create teams of 10
from the total number
of soccer players?
(2 points)
2.768148789∗1022

How are
odds and
probability
different?
(2 points)
Probability of an event occurring is the fraction
of the times the event is expected over the total
number of trials. Probability value ranges
between 0 and 1. On the other hand odds are
defined as the probability of an event occurring
divided by the probability of an event not taking
place.
What is the
odds in favor
ratio?
(3 points)
The odds in favor ratio is the fraction of number
of ways an outcome occurs over the number of
ways it cannot occur.
What is the
probability
of an event
ratio?
(3 points)
This is the ratio of a number of times an even
can occur over the total outcomes.
What are the odds of
rolling a three (use the
proper ratio from
above)? Simplify all
fraction answers.
(2 points)
Probability of rolling three 1
6
Complement 5
6
Odds of rolling three
1
6
5
6
= 1
5

b. Reflect on the previous question’s answer outcome. First, convert the fraction to a
percent.
Percent Probability
Theoretical
Probability (Rounded
to the nearest whole
percent.)
(2 points)
Converting probability to percent
1
6∗100=16.66 %
Which is 1 7 %
What is the theoretical
probability of rolling a
three (use the proper
ratio from above)?
Simplify all fraction
answers.
(2 points)
Number of events is 6
So, the probability is 1
6

Next, use the likelihood scale table above to select the term that best describes your
answer.
Likelihood Scale
Term
(2 points)
Unlikely
c. What if someone challenged you to never roll a 3? If you were to roll your single
six-sided die 18 times, what would be the empirical probability of never getting a
three?
Also, list the likelihood scale term from the table above.
Percent Probability
Solution
(Rounded to the
Nearest Whole
Percent)
(2 points)
0 %
Likelihood Scale
Term
(2 points)
Rare
Show your work here: (2 points)
The probability of an event never occurring is 0
d. After 18 rolls, what would be the empirical probability of getting a three on at
least one of those rolls? Also, list the likelihood scale term from the table above.
Percent Probability
Empirical
6 %

Probability
(Rounded to the
Nearest Whole
Percent)
(2 points)
Likelihood Scale
Term
(2 points)
Rare
Show your work: (2 points)
The 18 rolls are independent of each other hence the probability of getting a 3 on at
least one of the rolls is 1
18 =0.0556=5.55%=6 %
e. Roll
the die
18 times and keep track of what is rolled in the table below. Remember, if you do
not have one, you can use virtual dice: https://www.random.org/dice/ (2 points)
What do you notice
about the answers for
parts c and d above?
(2 points)
In a situation where an event can
never occur the probability of the
event is 9. On the other hand, when
the even can occur at least once, the
probability of obtained as 6%
indicating that such situations can
occur but are rare

Roll # Dice Roll # Dice Roll # Dice
Roll 1 1 Roll 7 1 Roll 13 1
Roll 2 3 Roll 8 6 Roll 14 4
Roll 3 6 Roll 9 2 Roll 15 5
Roll 4 3 Roll 10 4 Roll 16 2
Roll 5 2 Roll 11 3 Roll 17 4
Roll 6 3 Roll 12 3 Roll 18 2
f. Based on your die rolls, what is the experimental probability of rolling a three, out
of 18 rolls? Also, list the likelihood scale term from the table above.
Percent Probability
Experimental
Probability
(Rounded to the
Nearest Whole
Percent)
(2 points)
28 %
Likelihood Scale
Term
(2 points)
Unlikely