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Discrete Mathematics: An Open Introduction

Oscar Levin

Chapter 1

Counting - all with Video Answers

Educators

Section 1

Additive and Multiplicative Principles

01:32

Problem 1

Your wardrobe consists of 5 shirts, 3 pairs of pants, and 17 bow ties. bow ties How many different outfits can you make?

Gregory Higby
Gregory Higby
Numerade Educator
01:10

Problem 2

For your college interview, you must wear a tie. You own 3 regular (boring) ties and 5 (cool) bow ties.
(a) How many choices do you have for your neck-wear?
(b) You realize that the interview is for clown college, so you should probably wear both a regular tie and a bow tie. How many choices do you have now?
(c) For the rest of your outfit, you have 5 shirts, 4 skirts, 3 pants, and 7 dresses. You want to select either a shirt to wear with a skirt or pants, or just a dress. How many outfits do you have to choose from?

Narayan Hari
Narayan Hari
Numerade Educator
00:14

Problem 3

Your Blu-ray collection consists of 9 comedies and 7 horror movies. Give an example of a question for which the answer is:
(a) 16 .
(b) 63 .

Amy Jiang
Amy Jiang
Numerade Educator
01:14

Problem 4

hexadecimal We usually write numbers in decimal form (or base 10), meaning numbers are composed using 10 different "digits" $\{0,1, \ldots, 9\}$. Sometimes though it is useful to write numbers hexadecimal or base
16. Now there are 16 distinct digits that can be used to form numbers:
$\{0,1, \ldots, 9, \mathrm{~A}, \mathrm{~B}, \mathrm{C}, \mathrm{D}, \mathrm{E}, \mathrm{F}\} .$ So for example, a 3 digit hexadecimal
number might be $2 \mathrm{~B} 8$.
(a) How many 2 -digit hexadecimals are there in which the first digit is $E$ or $F$ ? Explain your answer in terms of the additive principle (using either events or sets).
(b) Explain why your answer to the previous part is correct in terms of the multiplicative principle (using either events or sets). Why do both the additive and multiplicative principles give you the same answer?
(c) How many 3 -digit hexadecimals start with a letter (A-F) and end with a numeral $(0-9) ?$ Explain.
(d) How many 3 -digit hexadecimals start with a letter (A-F) or end with a numeral $(0-9)$ (or both)? Explain.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:59

Problem 5

Suppose you have sets $A$ and $B$ with $|A|=10$ and $|B|=15$.
(a) What is the largest possible value for $|A \cap B|$ ?
(b) What is the smallest possible value for $|A \cap B|$ ?
(c) What are the possible values for $|A \cup B|$ ?

Nidhi Singhi
Nidhi Singhi
Numerade Educator
02:51

Problem 6

If $|A|=8$ and $|B|=5,$ what is $|A \cup B|+|A \cap B| ?$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
03:42

Problem 7

A group of college students were asked about their TV watching habits. Of those surveyed, 28 students watch The Walking Dead, 19 watch The Blacklist, and 24 watch Game of Thrones. Additionally, 16 watch The Walking Dead and The Blacklist, 14 watch The Walking Dead and Game of Thrones, and 10 watch The Blacklist and Game of Thrones. There are 8 students who watch all three shows. How many students surveyed watched at least one of the shows?

Nick Johnson
Nick Johnson
Numerade Educator
00:47

Problem 8

In a recent survey, 30 students reported whether they liked their potatoes Mashed, French-fried, or Twice-baked. 15 liked them mashed, 20 liked French fries, and 9 liked twice baked potatoes. Additionally, 12 students liked both mashed and fried potatoes, 5 liked French fries and twice baked potatoes, 6 liked mashed and baked, and 3 liked all three styles. How many students hate potatoes? Explain why your answer is correct.

WZ
Wen Zheng
Numerade Educator
02:29

Problem 9

For how many $n \in\{1,2, \ldots, 500\}$ is $n$ a multiple of one or more of 5 , $6,$ or $7 ?$

Jen H
Jen H
Numerade Educator
01:11

Problem 10

How many positive integers less than 1000 are multiples of $3,5,$ or $7 ?$ Explain your answer using the Principle of Inclusion/Exclusion.

Norman Atentar
Norman Atentar
Numerade Educator
01:19

Problem 11

Let $A, B,$ and $C$ be sets.
(a) Find $|(A \cup C) \backslash B|$ provided $|A|=50,|B|=45,|C|=40,|A \cap B|=$
$20,|A \cap C|=15,|B \cap C|=23,$ and $|A \cap B \cap C|=12$
(b) Describe a set in terms of $A, B,$ and $C$ with cardinality $26 .$

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
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Problem 12

Consider all 5 letter "words" made from the letters $a$ through $h$. (Recall, words are just strings of letters, not necessarily actual English words.)
(a) How many of these words are there total?
(b) How many of these words contain no repeated letters?
(c) How many of these words start with the sub-word "aha"?
(d) How many of these words either start with "aha" or end with "bah" or both?
(e) How many of the words containing no repeats also do not contain the sub-word "bad"?

Donna Densmore
Donna Densmore
Numerade Educator
01:59

Problem 13

For how many three digit numbers $(100$ to 999$)$ is the sum of the digits even? (For example, 343 has an even sum of digits: $3+4+3=10$ which is even.) Find the answer and explain why it is correct in at least two different ways.

Victor Salazar
Victor Salazar
Numerade Educator
00:14

Problem 14

The number 735000 factors as $2^{3} \cdot 3 \cdot 5^{4} \cdot 7^{2}$. How many divisors does it have? Explain your answer using the multiplicative principle.

Amy Jiang
Amy Jiang
Numerade Educator