Combinatorics

The Joy of a Teacher is the Success of his Students. - Samuel Dominic Chukwuemeka


Welcome to Combinatorics

Samdom For Peace

I greet you this day,
First: read the notes. Second: view the videos. Third: solve the questions/solved examples. Fourth: check your solutions with my thoroughly-explained solutions. Fifth: check your answers with the calculators as applicable.
Comments, ideas, areas of improvement, questions, and constructive criticisms are welcome. You may contact me.
If you are my student, please do not contact me here. Contact me via the school's system. Thank you for visiting!!!

Samuel Dominic Chukwuemeka (Samdom For Peace) B.Eng., A.A.T, M.Ed., M.S

Combinatorics


Objectives

Students will:
(1.) Discuss the basic concepts used in Combinatorics.
(2.) State the Fundamental Counting Principle.
(3.) Determine the number of ways two tasks can be done consecutively.
(4.) Determine the number of ways multiple tasks can be done successively.
(5.) Determine the number of permutations of items.
(6.) Determine the number of permutations of duplicate items.
(7.) Determine the number of permutations of total items taking some items at a time.
(8.) Determine the number of permutations of total items taking some items at a time.


Definition

Combinatorics is the mathematics of counting.
It is the branch of mathematics that deals with the counting of finite items, and the arrangement of finite items with or without regard to the order of arrangement.

Formulas


Say:

$n$ is the number of items ($n$ items)

$c$ and $d$ are the number of duplicate items

$n!$ is read as $n-factorials$

The number of permutations of $n$ items is $n!$

The number of permutations of duplicate items is $\dfrac{n!}{c! * d!}$

The number of permutations of $n$ total items taking $r$ items at a time is $^nP_r$ or $_nP_r$ or $P(n, r)$

The number of combinations of $n$ total items taking $r$ items at a time is $^nC_r$ or $_nC_r$ or $C(n, r)$

$ (1.)\:\: n! = n * (n - 1) * (n - 2) * (n - 3) * ... * 1 \\[3ex] (2.)\:\: P(n, r) = \dfrac{n!}{(n - r)!} \\[5ex] (3.)\:\: C(n, r) = \dfrac{n!}{(n - r)!r!} \\[5ex] $

Fundamental Counting Principle


The Fundamental Counting Principle is also referred to as:

The Fundamental Principle of Counting

OR
The Multiplication Principle

OR
The Counting Principle

OR
The Principle of Counting.


I sing ☺☺☺

I sing tenor.
I sing bass.

I dance ☺☺☺

I dance Davidic dance.
I dance Ibo traditional dance.

In how many ways can I sing and dance?

Let tenor be $T$
bass be $B$
Davidic dance be $D$
Ibo traditional dance = $I$

So, I can:
$T - D$ - Sing tenor and dance Davidic dance
$T - I$ - Sing tenor and dance Ibo traditional dance
$B - D$ - Sing bass and dance Davidic dance
$B - I$ - Sing bass and dance Ibo traditional dance

I can sing and dance in $4$ ways.
I can sing and dance in $(2 * 2)$ ways.
This leads us to...

The Fundamental Counting Principle states that if:
A task can be done in $A$ ways and
Another task can be done in $B$ ways;
then both tasks can be done consecutively in $A * B$ ways.

Student: What if we have to do multiple tasks?
Teacher: Good question!
That leads us to...

The Generalized Multiplication Principle states that the total number of ways of doing multiple tasks in succession is the product of the number of ways of doing each task individually.

Solved Examples


Questions and Answers
Specify the type of case for each question as applicable.

(1.) Mr. C has two shirts - a black shirt and a white shirt.
He has three pants - a black pant, a red pant, and a green pant.
In how many ways can he dress up for work using any combination of shirt and pant?
Show the combinations.


This is a case of the Fundamental Counting Principle
$2$ shirts
$3$ pants
The number of ways he can dress for work is $2 * 3 = 6$ ways

Let black shirt = $BS$
white shirt = $WS$
black pant = $BP$
red pant = $RP$
green pant = $GP$

The combinations are:
$BS - BP$
$BS - RP$
$BS - GP$
$WS - BP$
$WS - RP$
$WS - GP$

(2.) Zinne Diners offers ten main courses, three desserts, and seven sides.
How many ways can a person order a three-course meal? meal

Teacher: Who wants some "brain work"?
Or guess what...this could be a good punishment for children that misbehave
Student: Hmmm...what is it? Spanking?
Teacher: No...
The punishment would be to list all the combinations for the three course meal. ☺☺☺
Let main course = M, dessert = D, and side = S
So, you begin with: $M1D1S1, M1D1S2, M1D1S3,...$ and the list goes on till...
Student: $210$ lol...
I think it is better than spanking...much better


This is a case of the Fundamental Counting Principle
$10$ main courses
$3$ desserts
$7$ sides
Number of ways one can order a three-course meal = $10 * 3 * 7 = 210$ ways

(3.) ACT An automobile license plate number issued by a certain state has $6$ character positions.
Each of the first $3$ positions contains a single digit from $0$ through $9$.
Each of the last $3$ positions contains $1$ of the $26$ letters of the alphabet.
Digits and letters of the alphabet can be repeated on a license plate.
How many different such license plate numbers can be made?


This is a case of the Fundamental Counting Principle
There are ten digits from $0 - 9$
$0, 1, 2, 3, 4, 5, 6, 7, 8, 9$
There are twenty six letters from $A - Z$

Based on the question:
Any of the $10$ digits ($0 - 9$ is ten digits) can be placed in the first position.
Any of the $10$ digits ($0 - 9$ is ten digits) can be placed in the second position.
Any of the $10$ digits ($0 - 9$ is ten digits) can be placed in the third position.
Any of the $26$ letters ($A - Z$ is twenty six letters) can be placed in the fourth position.
Any of the $26$ letters ($A - Z$ is twenty six letters) can be placed in the fifth position.
Any of the $26$ letters ($A - Z$ is twenty six letters) can be placed in the sixth position.
This is seen as:
$\underline{0 - 9}$ $\underline{0 - 9}$ $\underline{0 - 9}$ $\underline{A - Z}$ $\underline{A - Z}$ $\underline{A - Z}$
$\underline{10}$ $\underline{10}$ $\underline{10}$ $\underline{26}$ $\underline{26}$ $\underline{26}$
Number of different license plates = $10 * 10 * 10 * 26 * 26 * 26 = 17,576,000$ license plates

(4.) socialsecurity The United States social security number (SSN) contains nine digits.
How many different social security numbers are possible if:
(a.) repetition of digits are allowed?
(b.) repetition of digits are not allowed?
(c.) repetition of digits are allowed and the first number cannot begin with a $0$?
(d.) repetition of digits are not allowed and the first number cannot begin with a $0$?


This is a case of the Fundamental Counting Principle
There are ten digits from $0 - 9$
$0, 1, 2, 3, 4, 5, 6, 7, 8, 9$

(a.) repetition of digits are allowed.
Any of the $10$ digits ($0 - 9$ is ten digits) can be placed in the any of the nine positions.
$\underline{0 - 9}$ $\underline{0 - 9}$ $\underline{0 - 9}$ $\underline{0 - 9}$ $\underline{0 - 9}$ $\underline{0 - 9}$ $\underline{0 - 9}$ $\underline{0 - 9}$ $\underline{0 - 9}$
$\underline{10}$ $\underline{10}$ $\underline{10}$ $\underline{10}$ $\underline{10}$ $\underline{10}$ $\underline{10}$ $\underline{10}$ $\underline{10}$
Number of social security numbers = $10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 = 1,000,000,000$ social security numbers.

(b.) repetition of digits are not allowed.
This means that once you use a digit, you cannot use it again.
So, any of the $10$ digits ($0 - 9$ is ten digits) can be placed in the first position.
Once you use that first digit, you cannot use it again.
Only $9$ digits are left.
Any of the $9$ digits can be placed in the second position.
Any of the $8$ digits can be placed in the third position.
Any of the $7$ digits can be placed in the fourth position.
Any of the $6$ digits can be placed in the fifth position.
Any of the $5$ digits can be placed in the sixth position.
Any of the $4$ digits can be placed in the seventh position.
Any of the $3$ digits can be placed in the eighth position.
Any of the $2$ digits can be placed in the ninth position.
$\underline{10}$ $\underline{9}$ $\underline{8}$ $\underline{7}$ $\underline{6}$ $\underline{5}$ $\underline{4}$ $\underline{3}$ $\underline{2}$
Number of social security numbers = $10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 = 3,628,800$ social security numbers.

(c.) repetition of digits are allowed and the first number cannot begin with a $0$.
Without the $0$, there are $9$ ($1 - 9$) digits
Any of the $9$ digits can be placed in the first position.
However, any of the $10$ digits can be placed in the second through ninth positions because the repetition of digits are allowed.
$\underline{9}$ $\underline{10}$ $\underline{10}$ $\underline{10}$ $\underline{10}$ $\underline{10}$ $\underline{10}$ $\underline{10}$ $\underline{10}$
Number of social security numbers = $9 * 10 * 10 * 10 * 10 * 10 * 10 * 10 * 10 = 900,000,000$ social security numbers.


(5.) Micah is taking a multiple choice question test that has $5$ questions and $4$ answer choices.
He must attempt all questions and select one choice for each question.
How many ways can he answer the questions?


This is a case of the Fundamental Counting Principle
$5$ questions _ Questions $1, 2, 3, 4, 5$
$4$ answer choices - Choices $A, B, C, D$
He can select any of the four answer choices for Question $1$
Any of the four answer choices can be chosen for Question $2$
Any of the four answer choices can be chosen for Question $3$
Any of the four answer choices can be chosen for Question $4$
Any of the four answer choices can be chosen for Question $5$
$\underline{A - D}$ $\underline{A - D}$ $\underline{A - D}$ $\underline{A - D}$ $\underline{A - D}$
$\underline{4}$ $\underline{4}$ $\underline{4}$ $\underline{4}$ $\underline{4}$
Number of ways the questions can be answered = $4 * 4 * 4 * 4 * 4 = 1024$ ways.

(6.) ACT In his costume supplies, Elmo the clown has $4$ noses, $3$ pairs of lips, and $2$ wigs.
A clown costume consists of $1$ nose, $1$ pair of lips, and $1$ wig.
How many different clown costumes can Elmo make?


This is a case of the Fundamental Counting Principle
$4$ noses
$3$ pairs of lips
$2$ wigs
Number of different clown costumes = $4 * 3 * 2 = 24$ costumes

(7.) phone ACT Get-A-Great-Read Books is adding a new phone line.
The phone company says that the first $3$ digits of the phone number must be $555$, but the remaining $4$ digits, where each digit is a digit from $0$ through $9$, can be chosen by Get-A-Great-Read Books.
How many phone numbers are possible?
$ A.\:\: 5(9^4) \\[3ex] B.\:\: 5^3(9^4) \\[3ex] C.\:\: 5^3(10^4) \\[3ex] D.\:\: 9^4 \\[3ex] E. 10^4 $


This is a case of the Fundamental Counting Principle
It is a $7-digit$ phone number
Only $1$ number, $(5)$ can be in the first position.
Only $1$ number, $(5)$ can be in the second position.
Only $1$ number, $(5)$ can be in the third position.
Any of the $10$ digits ($0 - 9$ is ten digits) can be placed in the fourth position.
Any of the $10$ digits ($0 - 9$ is ten digits) can be placed in the fifth position.
Any of the $10$ digits ($0 - 9$ is ten digits) can be placed in the sixth position.
Any of the $10$ digits ($0 - 9$ is ten digits) can be placed in the seventh position.
This is seen as:
$\underline{5}$ $\underline{5}$ $\underline{5}$ $\underline{0 - 9}$ $\underline{0 - 9}$ $\underline{0 - 9}$ $\underline{0 - 9}$
$\underline{1}$ $\underline{1}$ $\underline{1}$ $\underline{10}$ $\underline{10}$ $\underline{10}$ $\underline{10}$
Number of possible phone numbers = $1 * 1 * 1 * 10 * 10 * 10 * 10 = 10^4 = 10,000$ phone numbers

(8.) The local ten-digit telephone numbers in the City of Truth or Consequences, New Mexico have $575$ as the area code.
How many different telephone numbers are possible in the City of "Speak the Truth or Face the Consequences", New Mexico?


This is a case of the Fundamental Counting Principle
The telephone number is a $10-digit$ number
Only $1$ number, $(5)$ can be in the first position.
Only $1$ number, $(7)$ can be in the second position.
Only $1$ number, $(5)$ can be in the third position.
Any of the $10$ digits ($0 - 9$ is ten digits) can be placed in the fourth position.
Any of the $10$ digits ($0 - 9$ is ten digits) can be placed in the fifth position.
Any of the $10$ digits ($0 - 9$ is ten digits) can be placed in the sixth position.
Any of the $10$ digits ($0 - 9$ is ten digits) can be placed in the seventh position.
Any of the $10$ digits ($0 - 9$ is ten digits) can be placed in the eighth position.
Any of the $10$ digits ($0 - 9$ is ten digits) can be placed in the ninth position.
Any of the $10$ digits ($0 - 9$ is ten digits) can be placed in the tenth position.
This is seen as:
$\underline{5}$ $\underline{7}$ $\underline{5}$ $\underline{0 - 9}$ $\underline{0 - 9}$ $\underline{0 - 9}$ $\underline{0 - 9}$ $\underline{0 - 9}$ $\underline{0 - 9}$ $\underline{0 - 9}$
$\underline{1}$ $\underline{1}$ $\underline{1}$ $\underline{10}$ $\underline{10}$ $\underline{10}$ $\underline{10}$ $\underline{10}$ $\underline{10}$ $\underline{10}$
Number of possible phone numbers = $1 * 1 * 1 * 10 * 10 * 10 * 10 * 10 * 10 * 10 = 10^7 = 10,000,000$ phone numbers

(9.) stocks Sometimes, the value of a stock may go up, go down, or remain constant.
How many possibilities are there for someone who owns ten stocks?


This is a case of the Fundamental Counting Principle
$10$ stocks _ Stocks $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$
$3$ options - go up, go down, or remain constant

Stock $1$ may go up, go down, or remain constant
Stock $2$ may go up, go down, or remain constant
Stock $3$ may go up, go down, or remain constant
Stock $4$ may go up, go down, or remain constant
Stock $5$ may go up, go down, or remain constant
Stock $6$ may go up, go down, or remain constant
Stock $7$ may go up, go down, or remain constant
Stock $8$ may go up, go down, or remain constant
Stock $9$ may go up, go down, or remain constant
Stock $10$ may go up, go down, or remain constant
This is seen as:
$\underline{3}$ $\underline{3}$ $\underline{3}$ $\underline{3}$ $\underline{3}$ $\underline{3}$ $\underline{3}$ $\underline{3}$ $\underline{3}$ $\underline{3}$
Number of possibilities for ten stocks = $3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 * 3 = 3^{10} = 59049$ possibilities.

(10.) In the original plan for area codes in $1945$, the first digit could be any number from $2$ through $9$; the second digit was either $0$ or $1$; and the third digit could be any number except $0$.
How many different area codes are possible with this plan?


This is a case of the Fundamental Counting Principle
The area code is a $3-digit$ code
Any of the $8$ digits ($2 - 9$ is eight digits) can be the first digit.
Any of the $2$ digits ($0$ or $1$ is two digits) can be the second digit.
Any of the $9$ digits ($1 - 9$ is nine digits) can be the third digit.
This is seen as:
$\underline{2 - 9}$ $\underline{0 \:\:OR\:\: 1}$ $\underline{1 - 9}$
$\underline{8}$ $\underline{2}$ $\underline{9}$
Number of possible area codes = $8 * 2 * 9 = 144$ area codes

(11.) licenseplateGA The new license plate of the State of Georgia has three letters followed by five numbers.
How many license plates of this kind are possible if:
(a.) repetition of letters and numbers are allowed?
(b.) repetition of letters and numbers are not allowed?
(c.) repetition of letters are allowed but the repetition of numbers are not allowed?
(d.) repetition of letters are not allowed but the repetition of numbers are allowed?


This is a case of the Fundamental Counting Principle
There are ten digits from $0 - 9$
$0, 1, 2, 3, 4, 5, 6, 7, 8, 9$
There are twenty six letters from $A - Z$

(a.) repetition of letters and numbers are allowed?
Any of the $26$ letters ($A - Z$ is twenty six letters) can be placed in the first position.
Any of the $26$ letters ($A - Z$ is twenty six letters) can be placed in the second position.
Any of the $26$ letters ($A - Z$ is twenty six letters) can be placed in the third position.
Any of the $10$ digits ($0 - 9$ is ten digits) can be placed in the fourth position.
Any of the $10$ digits ($0 - 9$ is ten digits) can be placed in the fifth position.
Any of the $10$ digits ($0 - 9$ is ten digits) can be placed in the sixth position.
Any of the $10$ digits ($0 - 9$ is ten digits) can be placed in the seventh position.
Any of the $10$ digits ($0 - 9$ is ten digits) can be placed in the eighth position.
This is seen as:
$\underline{A - Z}$ $\underline{A - Z}$ $\underline{A - Z}$ $\underline{0 - 9}$ $\underline{0 - 9}$ $\underline{0 - 9}$ $\underline{0 - 9}$ $\underline{0 - 9}$
$\underline{26}$ $\underline{26}$ $\underline{26}$ $\underline{10}$ $\underline{10}$ $\underline{10}$ $\underline{10}$ $\underline{10}$
Number of license plates = $26 * 26 * 26 * 10 * 10 * 10 * 10 * 10 = 1,757,600,000$ license plates.

(b.) repetition of letters and numbers are not allowed?
This means that once you use a letter, you cannot use it again.
Also; once you use a digit/number, you cannot use it again.
Any of the $26$ letters ($A - Z$ is twenty six letters) can be placed in the first position.
Any of the $25$ letters can be placed in the second position.
Any of the $24$ letters can be placed in the third position.
Any of the $10$ digits ($0 - 9$ is ten digits) can be placed in the fourth position.
Any of the $9$ digits can be placed in the fifth position.
Any of the $8$ digits can be placed in the sixth position.
Any of the $7$ digits can be placed in the seventh position.
Any of the $6$ digits can be placed in the eighth position.
This is seen as:
$\underline{26}$ $\underline{25}$ $\underline{24}$ $\underline{10}$ $\underline{9}$ $\underline{8}$ $\underline{7}$ $\underline{6}$
Number of license plates = $26 * 25 * 24 * 10 * 9 * 8 * 7 * 6 = 471,744,000$ license plates.

(c.) repetition of letters are allowed but the repetition of numbers are not allowed?
This means that you can reuse the letters.
However; once you use a digit/number, you cannot use it again.
Any of the $26$ letters ($A - Z$ is twenty six letters) can be placed in the first position.
Any of the $26$ letters ($A - Z$ is twenty six letters) can be placed in the second position.
Any of the $26$ letters ($A - Z$ is twenty six letters) can be placed in the third position.
Any of the $10$ digits ($0 - 9$ is ten digits) can be placed in the fourth position.
Any of the $9$ digits can be placed in the fifth position.
Any of the $8$ digits can be placed in the sixth position.
Any of the $7$ digits can be placed in the seventh position.
Any of the $6$ digits can be placed in the eighth position.
This is seen as:
$\underline{26}$ $\underline{26}$ $\underline{26}$ $\underline{10}$ $\underline{9}$ $\underline{8}$ $\underline{7}$ $\underline{6}$
Number of license plates = $26 * 26 * 26 * 10 * 9 * 8 * 7 * 6 = 531,498,240$ license plates.

(d.) repetition of letters are not allowed but the repetition of numbers are allowed? This means that you cannot reuse the letters.
But, you can reuse the digits/numbers.
Any of the $26$ letters ($A - Z$ is twenty six letters) can be placed in the first position.
Any of the $25$ letters can be placed in the second position.
Any of the $24$ letters can be placed in the third position.
Any of the $10$ digits ($0 - 9$ is ten digits) can be placed in the fourth position.
Any of the $10$ digits ($0 - 9$ is ten digits) can be placed in the fifth position.
Any of the $10$ digits ($0 - 9$ is ten digits) can be placed in the sixth position.
Any of the $10$ digits ($0 - 9$ is ten digits) can be placed in the seventh position.
Any of the $10$ digits ($0 - 9$ is ten digits) can be placed in the eighth position.
This is seen as:
$\underline{26}$ $\underline{25}$ $\underline{24}$ $\underline{10}$ $\underline{10}$ $\underline{10}$ $\underline{10}$ $\underline{10}$
Number of license plates = $26 * 25 * 24 * 10 * 10 * 10 * 10 * 10 = 1,560,000,000$ license plates.


(12.) ACT A committee will be selected from a group of $12$ women and $18$ men.
The committee will consist of $5$ women and $5$ men.
Which of the following expressions gives the number of different committees that could be selected from these $30$ people?
$ F.\:\: _{30}P_{10} \\[3ex] G.\:\: (_{12}P_5)(_{18}P_5) \\[3ex] H.\:\: _{30}C_{10} \\[3ex] J.\:\: (_{30}C_5)(_{30}C_5) \\[3ex] K.\:\: (_{12}C_5)(_{18}C_5) $


This is a case of Combinations
This is because the selection does not have to be ordered
The committee can be formed by selecting "any $5$" women from $12$ women and "any $5$" men from $18$ men
Number of different committees that can be formed from this selection = $(_{12}C_5)(_{18}C_5)$

$ (_{12}C_5)(_{18}C_5) = C(12, 5) * C(18, 5) \\[3ex] C(n, r) = \dfrac{n!}{(n - r)!r!} \\[5ex] C(12, 5) = \dfrac{12!}{(12 - 5)!5!} \\[5ex] C(12, 5) = \dfrac{12!}{7!5!} \\[5ex] C(12, 5) = \dfrac{12!}{7!5!} \\[5ex] C(12, 5) = \dfrac{12 * 11 * 10 * 9 * 8 * 7!}{7! * 5 * 4 * 3 * 2 * 1} \\[5ex] C(12, 5) = 11 * 9 * 8 = 792 \\[3ex] C(18, 5) = \dfrac{18!}{(18 - 5)!5!} \\[5ex] C(18, 5) = \dfrac{18!}{13!5!} \\[5ex] C(18, 5) = \dfrac{18 * 17 * 16 * 15 * 14 * 13!}{13! * 5 * 4 * 3 * 2 * 1} \\[5ex] C(18, 5) = 18 * 17 * 2 * 14 = 8568 \\[3ex] $ Number of committees = $792 * 8568 = 6,785,856$ committees

(13.) callLetters How many different five-letter radio station call letters can be formed if the first letter must be $S$ or $C$?


This is a case of the Fundamental Counting Principle
This is a $5-lettered$ code
Any of the $2$ letters ($S$ or $C$) can be the first letter.
Any of the $26$ letters ($A - Z$ is twenty six letters) can be the second letter.
Any of the $26$ letters ($A - Z$ is twenty six letters) can be the third letter.
Any of the $26$ letters ($A - Z$ is twenty six letters) can be the fourth letter.
Any of the $26$ letters ($A - Z$ is twenty six letters) can be the fifth letter.
This is seen as:
$\underline{S \:\:OR\:\: C}$ $\underline{A - Z}$ $\underline{A - Z}$ $\underline{A - Z}$ $\underline{A - Z}$
$\underline{2}$ $\underline{26}$ $\underline{26}$ $\underline{26}$ $\underline{26}$
Number of radio station call letters = $2 * 26 * 26 * 26 * 26 = 913,952$ call letters

(14.) ACT The employees at a hotel reservation center assign an $8-digit$ confirmation number ($CN$) to each customer making a reservation.
The first digit in each $CN$ is $8$.
The other $7$ digits can be any digit $0$ through $9$, and digits may repeat.
How many possible $8-digit\:\: CNs$ are there?
$ A.\:\: 8^7 \\[3ex] B.\:\: 9^7 \\[3ex] C.\:\: 10^7 \\[3ex] D.\:\: 8^8 \\[3ex] E.\:\: 10^8 $


This is a case of the Fundamental Counting Principle
The confirmation number is an $8-digit$ number
Only $1$ number, $(8)$ can be the first digit.
Any of the $10$ digits ($0 - 9$ is ten digits) can be the second digit.
Any of the $10$ digits ($0 - 9$ is ten digits) can be the third digit.
Any of the $10$ digits ($0 - 9$ is ten digits) can be the fourth digit.
Any of the $10$ digits ($0 - 9$ is ten digits) can be the fifth digit.
Any of the $10$ digits ($0 - 9$ is ten digits) can be the sixth digit.
Any of the $10$ digits ($0 - 9$ is ten digits) can be the seventh digit.
Any of the $10$ digits ($0 - 9$ is ten digits) can be the eighth digit.
This is seen as:
$\underline{8}$ $\underline{0 - 9}$ $\underline{0 - 9}$ $\underline{0 - 9}$ $\underline{0 - 9}$ $\underline{0 - 9}$ $\underline{0 - 9}$ $\underline{0 - 9}$
$\underline{1}$ $\underline{10}$ $\underline{10}$ $\underline{10}$ $\underline{10}$ $\underline{10}$ $\underline{10}$ $\underline{10}$
Number of possible confirmation numbers = $1 * 10 * 10 * 10 * 10 * 10 * 10 * 10 = 10^7 = 10,000,000$ confirmation numbers

(15.) $C-Mart$ Stores has a new store in Two Egg, Florida.
One of their employees has to paint the parking spaces with a letter of the alphabet and a single digit from $1$ to $9$.
The first parking space is $A1$ and the last parking space is $Z9$.
How many parking spaces can be painted with distinct labels?


This is a case of the Fundamental Counting Principle
There are nine digits from $1 - 9$
$1, 2, 3, 4, 5, 6, 7, 8, 9$
There are twenty six letters from $A - Z$

Based on the question:
Any of the $26$ letters ($A - Z$ is twenty six letters) can be written in the first position.
Any of the $9$ digits ($1 - 9$ is nine digits) can be written in the second position.
This is seen as:
$\underline{A - Z}$ $\underline{1 - 9}$
$\underline{26}$ $\underline{9}$
Number of distinctly-marked parking spaces = $26 * 9 = 234$ parking spaces.

(16.) Find the number of permutations of the word, SAMUEL


This is a case of Permutation
SAMUEL has $6$ letters
Number of permutations of SAMUEL = $6! = 6 * 5 * 4 * 3 * 2 * 1 = 720$ permutations

Would it be a good punishment to list all the permutations of the word, SAMUEL?
SAMUEL
SAMULE
SAMEUL
SAMELU...up to $720$ of them
Well, it's better that spanking ☺☺☺

(17.) Find the number of permutations of the word, CHUKWUEMEKA


This is a case of Permutation of Duplicates
CHUKWUEMEKA has:
$11$ letters
$2$ U's
$2$ K's
$2$ W's
Number of permutations of CHUKWUEMEKA = $\dfrac{11!}{2! * 2! * 2!} = 4989600$ permutations

(18.) Find the number of permutations of the word, MATHEMATICS


This is a case of Permutation of Duplicates
MATHEMATICS has:
$11$ letters
$2$ M's
$2$ A's
$2$ T's
Number of permutations of MATHEMATICS = $\dfrac{11!}{2! * 2! * 2!} = 4989600$ permutations

(19.) In how many ways can the digits in the number $345237573$ be arranged?


This is a case of Permutation of Duplicates
$345237573$ has:
$9$ digits
$3$ 3's
$2$ 5's
$2$ 7's
Number of permutations of MATHEMATICS = $\dfrac{9!}{3! * 2! * 2!} = 15120$ ways

(20.) In how many ways can seven people line up at Register $3$ in a certain Walmart store to check out?


This is a case of Permutation
Number of ways = $7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5040$ ways

It is also a case of the Fundamental Counting Principle
Any of the $7$ people can stand in the first spot (spot closest to the cash register).
Once this is done, any of the remaining $6$ people can stand in the second spot.
Any of the remaining $5$ people can stand in the third spot.
Any of the remaining $4$ people can stand in the fourth spot.
Any of the remaining $3$ people can stand in the fifth spot.
Any of the remaining $2$ people can stand in the sixth spot.
The last person can stand in the seventh spot (the spot farthest from the cash register) in this scenario
This is seen as:
$\underline{7}$ $\underline{6}$ $\underline{5}$ $\underline{4}$ $\underline{3}$ $\underline{2}$ $\underline{1}$
Number of radio station call letters = $7 * 6 * 5 * 4 * 3 * 2 * 1 = 5040$ ways

(21.) ACT The positive integer $n!$ is defined as the product of all the positive integers less than or equal to $n$.
For example, $3! = 1(2)(3) = 6$.
What is the value of the expression $\dfrac{6!}{3!2!}$?


$ \dfrac{6!}{3!2!} \\[5ex] = \dfrac{1(2)(3)(4)(5)(6)}{(1)(2)(3) * (1)(2)} \\[5ex] = 1 * 3 * 4 * 5 \\[3ex] = 60 $

References


Chukwuemeka, S.D (2016, April 30). Samuel Chukwuemeka Tutorials - Math, Science, and Technology. Retrieved from https://www.samuelchukwuemeka.com
Blitzer, R. (2015). Thinking Mathematically ($6^{th}$ ed.). Boston: Pearson
Tan, S. (2015). Finite Mathematics for the Managerial, Life, and Social Sciences ($11^{th}$ ed.). Boston: Cengage Learning.
Triola, M. F. (2015). Elementary Statistics using the TI-83/84 Plus Calculator ($5^{th}$ ed.). Boston: Pearson
Georgia Gov. (n.d.). Retrieved from https://georgia.gov/blog/2012-05-16/new-license-plates-available
Microsoft Office Clip Art. (n.d.). Retrieved from https://support.office.com/en-us/article/add-clip-art-to-your-file-0a01ae25-973c-4c2c-8eaf-8c8e1f9ab530?legRedir=true&CorrelationId=9c8bb412-68e2-4db0-9704-c9af116e2521&ui=en-US&rs=en-US&ad=US