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Discrete Mathematics with Applications

Thomas Koshy

Chapter 6

Combinatorics and Discrete Probability - all with Video Answers

Educators

Section 1

The Fundamental Counting Principles

00:46

Problem 1

Find the number of positive integers $\leq 1976$ and divisible by:
$$
2 \text { or } 3
$$

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
06:34

Problem 2

Find the number of positive integers $\leq 1976$ and divisible by:
$$
3 \text { or } 5
$$

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
06:34

Problem 3

Find the number of positive integers $\leq 1976$ and divisible by:
$$
2,3, \text { or } 5
$$

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
04:30

Problem 4

Find the number of positive integers $\leq 1976$ and divisible by:
$$
3,5, \text { or } 7
$$

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
06:34

Problem 5

Find the number of positive integers $\leq 1976$ and divisible by:
In one version of BASIC, a variable name consists of a letter, or a letter followed by a digit, or the dollar sign (8). Find the the total number of possible variable names.

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
01:46

Problem 6

Find the number of terms in the expansion of each expression.
$$
(a+b)(c+d+e)(x+y)
$$

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
01:46

Problem 7

Find the number of terms in the expansion of each expression.
$$
(b+c)(d+e+f)(x+y+z)
$$

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
02:45

Problem 8

Find the number of terms in the expansion of each expression.
$$
\left(\sum_{i=0}^{2} a_{i}\right)\left(\sum_{i=1}^{4} b_{i}\right)\left(\sum_{i=2}^{5} c_{i}\right)
$$

AG
Ankit Gupta
Numerade Educator
02:45

Problem 9

Find the number of terms in the expansion of each expression.
$$
\left(\sum_{i=-2}^{5} a_{i}\right)\left(\sum_{i=-1}^{3} b_{i}\right)\left(\sum_{i=0}^{4} c_{i}\right)
$$

AG
Ankit Gupta
Numerade Educator
View

Problem 10

Find the number of palindromes of length $n$ over the English alphabet.

Claire Rochford
Claire Rochford
Numerade Educator
View

Problem 11

Find the number of palindromic alphanumeric identifiers of length $n$ . (See Example 6.7.)

Claire Rochford
Claire Rochford
Numerade Educator
View

Problem 12

Find the total number of bytes. (See Example $6.8 .$

Nick Johnson
Nick Johnson
Numerade Educator
00:45

Problem 13

A word over the alphabet $\{0,1,2\}$ is called a ternary word. Find the number of ternary words of length $n$ that can be formed.

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
01:05

Problem 14

A typical automobile license plate in New York contains three letters followed by three digits. Find the number of license plates of this kind that:
Can be formed.

Kyler Gray
Kyler Gray
Numerade Educator
00:41

Problem 15

A typical automobile license plate in New York contains three letters followed by three digits. Find the number of license plates of this kind that:
Begin with the word BAT.

Clarissa Noh
Clarissa Noh
Numerade Educator
00:41

Problem 16

A typical automobile license plate in New York contains three letters followed by three digits. Find the number of license plates of this kind that:
Begin with a vowel.

Clarissa Noh
Clarissa Noh
Numerade Educator
01:05

Problem 17

A typical automobile license plate in New York contains three letters followed by three digits. Find the number of license plates of this kind that:
Begin with the digit 6

Kyler Gray
Kyler Gray
Numerade Educator
02:58

Problem 18

A typical automobile license plate in New York contains three letters followed by three digits. Find the number of license plates of this kind that:
Repeat no letters or digits.

Xiaomeng Zhang
Xiaomeng Zhang
Numerade Educator
02:58

Problem 19

A typical automobile license plate in New York contains three letters followed by three digits. Find the number of license plates of this kind that:
Contain the same letters and the same digits.

Xiaomeng Zhang
Xiaomeng Zhang
Numerade Educator
02:58

Problem 20

A typical automobile license plate in New York contains three letters followed by three digits. Find the number of license plates of this kind that:
Have the property that both words and numbers are palindromes.

Xiaomeng Zhang
Xiaomeng Zhang
Numerade Educator
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Problem 21

An old zip code in the United States consists of five digits. Find the total number of possible zip codes that:
Have no repetitions.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
01:30

Problem 22

An old zip code in the United States consists of five digits. Find the total number of possible zip codes that:
Begin with 0

James Kiss
James Kiss
Numerade Educator
01:30

Problem 23

An old zip code in the United States consists of five digits. Find the total number of possible zip codes that:
End in K.

James Kiss
James Kiss
Numerade Educator
01:30

Problem 24

An old zip code in the United States consists of five digits. Find the total number of possible zip codes that:
Are palindromes.

James Kiss
James Kiss
Numerade Educator
00:59

Problem 25

A zip code in Canada consists of three letters and three digits. Each zip code begins with a letter. The letters and digits alternate; for instance, A1 B2 C3. Find the number of zip codes that:
End in 6

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:59

Problem 26

A zip code in Canada consists of three letters and three digits. Each zip code begins with a letter. The letters and digits alternate; for instance, A1 B2 C3. Find the number of zip codes that:
Begin with $\mathrm{A}$ and end in 3

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:59

Problem 27

A zip code in Canada consists of three letters and three digits. Each zip code begins with a letter. The letters and digits alternate; for instance, A1 B2 C3. Find the number of zip codes that:
End in $\mathrm{Z}$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:59

Problem 28

A zip code in Canada consists of three letters and three digits. Each zip code begins with a letter. The letters and digits alternate; for instance, A1 B2 C3. Find the number of zip codes that:
Are possible.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:49

Problem 29

The password for a computer system consists of six alphanumeric characters and begins with a letter. Find the total number of passwords that:
Are possible.

Maxime Rossetti
Maxime Rossetti
Numerade Educator
01:49

Problem 30

The password for a computer system consists of six alphanumeric characters and begins with a letter. Find the total number of passwords that:
End in RED.

Maxime Rossetti
Maxime Rossetti
Numerade Educator
01:49

Problem 31

The password for a computer system consists of six alphanumeric characters and begins with a letter. Find the total number of passwords that:
Contain the word BAT.

Maxime Rossetti
Maxime Rossetti
Numerade Educator
02:15

Problem 32

The password for a computer system consists of six alphanumeric characters and begins with a letter. Find the total number of passwords that:
End in 2076

Christopher Stanley
Christopher Stanley
Numerade Educator
03:58

Problem 33

Every radio and television station in the United States has a unique call name. Each contains three or four letters, beginning with $\mathrm{K}$ or $\mathrm{W}$ . For example, KEY and WASP are legal call names. Find the number of possible call names.

AG
Ankit Gupta
Numerade Educator
View

Problem 34

Find the number of bytes that:
Begin with 101

Nick Johnson
Nick Johnson
Numerade Educator
View

Problem 35

Find the number of bytes that:
End with 110

Nick Johnson
Nick Johnson
Numerade Educator
View

Problem 36

Find the number of bytes that:
Begin with and end in the same bit.

Nick Johnson
Nick Johnson
Numerade Educator
View

Problem 37

Find the number of bytes that:
Begin with and end in different bits.

Nick Johnson
Nick Johnson
Numerade Educator
01:01

Problem 38

Find the number of bytes that:
Have the same third and fourth bits.

Maxime Rossetti
Maxime Rossetti
Numerade Educator
01:06

Problem 39

Find the number of bytes that:
Have third bit or fifth bit 1.

Aman Gupta
Aman Gupta
Numerade Educator
01:06

Problem 40

Find the number of bytes that:
Have second bit 0 or third bit 1.

Aman Gupta
Aman Gupta
Numerade Educator
View

Problem 41

Find the number of bytes that:
Are palindromes.

Claire Rochford
Claire Rochford
Numerade Educator
00:58

Problem 42

Find the number of positive divisors of the following positive integers.
$2^{i} \cdot 3^{j} \cdot 5^{k}$

Hannah Laper
Hannah Laper
Numerade Educator
01:27

Problem 43

Find the number of positive divisors of the following positive integers.
$$
600
$$

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
01:04

Problem 44

A number-theoretic function used in the study of perfect numbers is
the tau function $\tau$ on $N .(\tau \text { is the Greek letter, tau.) } \tau(n) \text { denotes }$
the number of positive divisors of $n \in \mathbb{N} .$ Let $n=p_{1}^{\prime i} p_{2}^{c_{2}} \cdots p_{k}^{n_{k}},$ where $p_{1}, p_{2}, \ldots, p_{k}$ are distinct primes and $e_{1}, e_{2}, \ldots, e_{k} \in \mathbf{W} .$ Find $\tau(n)$

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
View

Problem 45

Find the number of $n \times n$ matrices that can be constructed using bits.

Nick Johnson
Nick Johnson
Numerade Educator
00:45

Problem 46

Find the number of ternary words that have:
Length at most 3

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
00:45

Problem 47

Find the number of ternary words that have:
Length at most 5 .

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
02:59

Problem 48

Find the number of ternary words that have:
Length at most $n$

Anthony Ramos
Anthony Ramos
Numerade Educator
View

Problem 49

Find the number of ternary words that have:
Length 3 and are palindromes.

Claire Rochford
Claire Rochford
Numerade Educator
View

Problem 50

Find the number of ternary words that have:
Length 4 and are palindromes.

Claire Rochford
Claire Rochford
Numerade Educator
00:45

Problem 51

Find the number of ternary words that have:
$4 \leq$ length $\leq 6$

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
00:45

Problem 52

In an alphabet of $m$ characters, how many words have:
Length 3?

Clarissa Noh
Clarissa Noh
Numerade Educator
00:45

Problem 53

In an alphabet of $m$ characters, how many words have:
Length not more than 2?

Clarissa Noh
Clarissa Noh
Numerade Educator
02:58

Problem 54

In an alphabet of $m$ characters, how many words have:
Length at least 2, but not more than 4?

Anthony Ramos
Anthony Ramos
Numerade Educator
00:45

Problem 55

In an alphabet of $m$ characters, how many words have:
Length not more than $n ?$

Clarissa Noh
Clarissa Noh
Numerade Educator
View

Problem 56

Let $A$ and $B$ be two finite sets with $|A|=m$ and $|B|=n .$ How many:
Functions can be defined from $A$ to $B ?$

Lauren Long
Lauren Long
Numerade Educator
01:43

Problem 57

Let $A$ and $B$ be two finite sets with $|A|=m$ and $|B|=n .$ How many:
Bijections can be defined from $A$ to $B$ (assume $m=n ) ?$

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
View

Problem 58

Let $A$ and $B$ be two finite sets with $|A|=m$ and $|B|=n .$ How many:
Invertible functions can be defined from $A$ to $B$ (assume $m=n ) ?$

Lauren Long
Lauren Long
Numerade Educator
01:43

Problem 59

Let $A$ and $B$ be two finite sets with $|A|=m$ and $|B|=n .$ How many:
Injections can be defined from $A$ to $B$ (assume $m \leq n ) ?$

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
02:00

Problem 60

Let $t$ denote the tau function. Prove each.
$t(n)$ is odd if and only if $n$ is a square.

Adriano Chikande
Adriano Chikande
Numerade Educator
03:56

Problem 61

Let $t$ denote the tau function. Prove each.
If $m$ and $n$ are relatively prime numbers, then $\tau(m n)=t(m) \cdot t(n)$

Sriparna Bhattacharjee
Sriparna Bhattacharjee
Numerade Educator
01:39

Problem 62

The harmonic mean $m$ of the numbers $a_{1}, a_{2}, \ldots, a_{n}$ is the reciprocal of the arithmetic mean of their reciprocals; that is,
$$\frac{1}{m}=\frac{1}{n} \sum_{i=1}^{n}\left(\frac{1}{a_{i}}\right)$$
Prove that the harmonic mean of the positive factors of a perfect number $N$ is an integer.
(Hint: If $d$ is a factor of $N$ , then so is $N / d .$ ) (R. Euler, 1987 )

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator