Introductory Combinatorics

Richard A. Brualdi

Chapter 2 Permutations and Combinations - all with Video Answers

Educators

Chapter Questions

Problem 1

For each of the four subsets of the two properties (a) and (b), count the number of four-digit numbers whose digits are either $1,2,3,4$, or $5:$
(a) The digits are distinct.
(b) The number is even.
Note that there are four problems here: $\emptyset$ (no further restriction), \{a\} (property
(a) holds), \{b\} (property
(b) holds), $\{a, b\}$ (both properties (a) and (b) hold).

Ankit Gupta

Numerade Educator

01:01

Problem 2

How many orderings are there for a deck of 52 cards if all the cards of the same suit are together?

Viraj Gaggar

Numerade Educator

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Problem 3

In how many ways can a poker hand (five cards) be dealt? How many different poker hands are there?

James Kiss

Numerade Educator

03:49

Problem 4

How many distinct positive divisors does each of the following numbers have?
(a) $3^{4} \times 5^{2} \times 7^{6} \times 11$
(b) 620
(c) $10^{10}$

Willis James

Numerade Educator

00:45

Problem 5

Determine the largest power of 10 that is a factor of the following numbers (equivalently, the number of terminal 0s, using ordinary base 10 representation):
(a) $50 !$
(b) $1000 !$

Aadit Sharma

Numerade Educator

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Problem 6

How many integers greater than 5400 have both of the following properties?
(a) The digits are distinct.
(b) The digits 2 and 7 do not occur.

Nick Johnson

Numerade Educator

05:16

Problem 7

In how many ways can four men and eight women be seated at a round table if there are to be two women between consecutive men around the table?

Gregory Higby

Numerade Educator

01:05

Problem 8

In how many ways can six men and six women be seated at a round table if the men and women are to sit in alternate seats?

Narayan Hari

Numerade Educator

02:55

Problem 9

In how many ways can 15 people be seated at a round table if $B$ refuses to sit next to A? What if $B$ only refuses to sit on A's right?

Vishal Sharma

Numerade Educator

01:36

Problem 10

A committee of five people is to be chosen from a club that boasts a membership of $10 \mathrm{men}$ and 12 women. How many ways can the committee be formed if it is to contain at least two women? How many ways if, in addition, one particular man and one particular woman who are members of the club refuse to serve together on the committee?

Clarissa Noh

Numerade Educator

01:31

Problem 11

How many sets of three integers between 1 and 20 are possible if no two consecutive integers are to be in a set?

Allison Knapp

Numerade Educator

01:39

Problem 12

A football team of 11 players is to be selected from a set of 15 players, 5 of whom can play only in the backfield, 8 of whom can play only on the line, and 2 of whom can play either in the backfield or on the line. Assuming a football team has 7 men on the line and 4 men in the backfield, determine the number of football teams possible.

Goutam Chand

Numerade Educator

03:44

Problem 13

There are 100 students at a school and three dormitories, $\mathrm{A}, \mathrm{B}$, and $\mathrm{C}$, with capacities 25, 35 and 40 , respectively.
(a) How many ways are there to fill the dormitories?
(b) Suppose that, of the 100 students, 50 are men and 50 are women and that A is an all-men's dorm, $B$ is an all-women's dorm, and $C$ is co-ed. How many ways are there to fill the dormitories?

Fasiha Binat Zafar

Numerade Educator

00:50

Problem 14

A classroom has two rows of eight seats each. There are 14 students, 5 of whom always sit in the front row and 4 of whom always sit in the back row. In how many ways can the students be seated?

Elizabeth Xu

Numerade Educator

01:36

Problem 15

At a party there are 15 men and 20 women.
(a) How many ways are there to form 15 couples consisting of one man and one woman?
(b) How many ways are there to form 10 couples consisting of one man and one woman?

Clarissa Noh

Numerade Educator

01:36

Problem 16

Prove that
$$
\left(\begin{array}{l}
n \\
r
\end{array}\right)=\left(\begin{array}{c}
n \\
n-r
\end{array}\right)
$$
by using a combinatorial argument and not the values of these numbers as given in Theorem 3.3.1.

Heather Zimmers

Numerade Educator

01:02

Problem 17

In how many ways can six indistinguishable rooks be placed on a 6 -by-6 board so that no two rooks can attack one another? In how many ways if there are two red and four blue rooks?

Aymara Gallardo

Numerade Educator

01:52

Problem 18

In how many ways can two red and four blue rooks be placed on an 8 -by-8 board so that no two rooks can attack one another?

Paul Gabriel

Numerade Educator

01:18

Problem 19

We are given eight, rooks, five of which are red and three of which are blue.
(a) In how many ways can the eight rooks be placed on an 8 -by- 8 chessboard so that no two rooks can attack one another?
(b) In how many ways can the eight rooks be placed on a 12 -by-12 chessboard so that no two rooks can attack one another?

Manik Pulyani

Numerade Educator

01:07

Problem 20

Determine the number of circular permutations of $\{0,1,2, \ldots, 9\}$ in which 0 and 9 are not opposite. (Hint: Count those in which 0 and 9 are opposite.)

Varsha Aggarwal

Numerade Educator

02:44

Problem 21

How many permutations are there of the letters of the word ADDRESSES? How many 8-permutations are there of these nine letters?

Pammi Eswari

Numerade Educator

02:48

Problem 22

A footrace takes place among four runners. If ties are allowed (even all four runners finishing at the same time), how many ways are there for the race to finish?

Shenade Gordon

Numerade Educator

01:49

Problem 23

Bridge is played with four players and an ordinary deck of 52 cards. Each player begins with a hand of 13 cards. In how many ways can a bridge game start? (Ignore the fact that bridge is played in partnerships.)

Aymara Gallardo

Numerade Educator

01:35

Problem 24

A roller coaster has five cars, each containing four seats, two in front and two in back. There are 20 people ready for a ride. In how many ways can the ride begin? What if a certain two people want to sit in different cars?

Sheryl Ezze

Numerade Educator

01:02

Problem 25

A ferris wheel has five cars, each containing four seats in a row. There are 20 people ready for a ride. In how many ways can the ride begin? What if a certain two people want to sit in different cars?

Sanchit Jain

Numerade Educator

02:02

Problem 26

A group of $m n$ people are to be arranged into $m$ teams each with $n$ players.
(a) Determine the number of ways if each team has a different name.
(b) Determine the number of ways if the teams don't have names.

Lucas Finney

Numerade Educator

01:34

Problem 27

In how many ways can five indistinguishable rooks be placed on an 8 -by-8 chessboard so that no rook can attack another and neither the first row nor the first column is empty?

Amany Waheeb

Numerade Educator

01:48

Problem 28

A secretary works in a building located nine blocks east and eight blocks north of his home. Every day he walks 17 blocks to work. (See the map that follows.)
(a) How many different routes are possible for him?
(b) How many different routes are possible if the one block in the easterly direction, which begins four blocks east and three blocks north of his home, is under water (and he can't swim)? (Hint Count the routes that use the block under water.)

Ankit Gupta

Numerade Educator

01:36

Problem 29

Let $S$ be a multiset with repetition numbers $n_{1}, n_{2}, \ldots, n_{k}$, where $n_{1}=1 .$ Let $n=n_{2}+\cdots+n_{k} .$ Prove that the number of circular permutations of $S$ equals
$$
\frac{n !}{n_{2} ! \cdots n_{k} !}
$$.

Heather Zimmers

Numerade Educator

01:03

Problem 30

We are to seat five boys, five girls, and one parent in a circular arrangement around a table. In how many ways can this be done if no boy is to sit next to a boy and no girl is to sit next to a girl? What if there are two parents?

Hast Aggarwal

Numerade Educator

01:18

Problem 31

In a soccer tournament of 15 teams, the top three teams are awarded gold, silver, and bronze cups, and the last three teams are dropped to a lower league. We regard two outcomes of the tournament as the same if the teams that receive the gold, silver, and bronze cups, respectively, are identical and the teams which drop to a lower league are also identical. How many different possible outcomes are there for the tournament?

Lauren Shelton

Numerade Educator

00:46

Problem 32

Determine the number of 11 -permutations of the multiset
$$
S=\{3 \cdot a, 4 \cdot b, 5 \cdot c\} .
$$

Ashley Volpe

Numerade Educator

00:46

Problem 33

Determine the number of 10-permutations of the multiset
$$
S=\{3 \cdot a, 4 \cdot b, 5 \cdot c\} .
$$

Ashley Volpe

Numerade Educator

01:13

Problem 34

Determine the number of 11 -permutations of the multiset
$$
S=\{3 \cdot a, 3 \cdot b, 3 \cdot c, 3 \cdot d\} .
$$

Vysakh M

Numerade Educator

04:52

Problem 35

List all 3-combinations and 4-combinations of the multiset
$$
\{2 \cdot a, 1 \cdot b, 3 \cdot c\}
$$

Aymara Gallardo

Numerade Educator

03:30

Problem 36

Determine the total number of combinations (of any size) of a multiset of objects of $k$ different types with finite repetition numbers $n_{1}, n_{2}, \ldots, n_{k}$, respectively.

Wen Zheng

Numerade Educator

00:56

Problem 37

A bakery sells six different kinds of pastry. If the bakery has at least a dozen of each kind, how many different options for a dozen of pastries are there? What if a box is to contain at least one of each kind of pastry?

Sheryl Ezze

Numerade Educator

04:36

Problem 38

How many integral solutions of
$$
x_{1}+x_{2}+x_{3}+x_{4}=30
$$
satisfy $x_{1} \geq 2, x_{2} \geq 0, x_{3} \geq-5$, and $x_{4} \geq 8 ?$

Muhammad Nawaz

Numerade Educator

02:34

Problem 39

There are 20 identical sticks lined up in a row occupying 20 distinct places as follows:
$|||||||||||||||||||| \mid .$
Six of them are to be chosen.
(a) How many choices are there?
(b) How many choices are there if no two of the chosen sticks can be consecutive?
(c) How many choices are there if there must be at least two sticks between each pair of chosen sticks?

Sheryl Ezze

Numerade Educator

01:18

Problem 40

There are $n$ sticks lined up in a row, and $k$ of them are to be chosen.
(a) How many choices are there?
(b) How many choices are there if no two of the chosen sticks can be consecutive?
(c) How many choices are there if there must be at least $l$ sticks between each pair of chosen sticks?

Manik Pulyani

Numerade Educator

01:36

Problem 41

In how many ways can 12 indistinguishable apples and 1 orange be distributed among three children in such a way that each child gets at least one piece of fruit?

Ashley Volpe

Numerade Educator

00:59

Problem 42

Determine the number of ways to distribute 10 orange drinks, 1 lemon drink, and 1 lime drink to four thirsty students so that each student gets at least one drink, and the lemon and lime drinks go to different students.

Hossam Mohamed

Numerade Educator

00:48

Problem 43

Determine the number of $r$ -combinations of the multiset
$$
\left\{1 \cdot a_{1}, \infty \cdot a_{2}, \ldots, \infty \cdot a_{k}\right\}
$$.

Kristof Mueller

Numerade Educator

02:31

Problem 44

Prove that the number of ways to distribute $n$ different objects among $k$ children equals $k^{n}$.

Wen Zheng

Numerade Educator

01:08

Problem 45

Twenty different books are to be put on five book shelves, each of which holds at least twenty books.
(a) How many different arrangements are there if you only care about the number of books on the shelves (and not which book is where)?
(b) How many different arrangements are there if you care about which books are where, but the order of the books on the shelves doesn't matter?
(c) How many different arrangements are there if the order on the shelves does matter?

Sneha Ravi

Numerade Educator

01:22

Problem 46

(a) There is an even number $2 n$ of people at a party, and they talk together in pairs, with everyone talking with someone (so $n$ pairs). In how many different ways can the $2 n$ people be talking like this?
(b) Now suppose that there is an odd number $2 n+1$ of people at the party with everyone but one person talking with someone. How many different pairings are there?

Aman Gupta

Numerade Educator

01:13

Problem 47

There are $2 n+1$ identical books to be put in a bookcase with three shelves. In how many ways can this be done if each pair of shelves together contains more books than the other shelf?

Kyler Gray

Numerade Educator

00:59

Problem 48

Prove that the number of permutations of $m A$ 's and at most $n B$ 's equals
$$
\left(\begin{array}{c}
m+n+1 \\
m+1
\end{array}\right)
$$.

Heather Zimmers

Numerade Educator

00:59

Problem 49

Prove that the number of permutations of at most $m A^{\prime} s$ and at most $n B^{\prime} s$ equals
$$
\left(\begin{array}{c}
m+n+2 \\
m+1
\end{array}\right)-1
$$.

Heather Zimmers

Numerade Educator

01:18

Problem 50

In how many ways can five identical rooks be placed on the squares of an 8 -by-8 board so that four of them form the corners of a rectangle with sides parallel to the sides of the board?

Manik Pulyani

Numerade Educator

01:31

Problem 51

Consider the multiset $\{n \cdot a, 1,2,3, \ldots, n\}$ of size $2 n$. Determine the number of its $n$ -combinations.

Manisha Sarker

Numerade Educator

04:47

Problem 52

Consider the multiset $\{n \cdot a, n \cdot b, 1,2,3, \ldots, n+1\}$ of size $3 n+1$. Determine the number of its $n$ -combinations.

Mohamed Mohamed

Numerade Educator

05:02

Problem 53

Find a one-to-one correspondence between the permutations of the set $\{1,2, \ldots, n\}$ and the towers $A_{0} \subset A_{1} \subset A_{2} \subset \cdots \subset A_{n}$ where $\left|A_{k}\right|=k$ for $k=0,1,2, \ldots, n$.

James Chok

Numerade Educator

02:45

Problem 54

Determine the number of towers of the form $\emptyset \subseteq A \subseteq B \subseteq\{1,2, \ldots, n\}$.

Mohamed Raafat Mohamed

Numerade Educator

01:58

Problem 55

How many permutations are there of the letters in the words
(a) TRISKAIDEKAPHOBIA (fear of the number 13$)$ ?
(b) FLOCCINAUCINIHILIPILIFICATION (estimating something as worthless)?
(c) PNEUMONOULTRAMICROSCOPICSILICOVOLCANOCONIOSIS (a lung disease caused by inhaling fine particles of silica)? (This word is, by some accounts, the longest word in the English language.)
(d) DERMATOGLYPHICS (skin patterns or the study of them)? (This word is the (current) longest word in the English language that doesn't repeat a letter; another word of the same length is UNCOPYRIGHTABLE. $^{13}$ )

Aman Gupta

Numerade Educator

01:30

Problem 56

What is the probability that a poker hand contains a flush (that is, five cards of the same suit)?

James Chok

Numerade Educator

01:27

Problem 57

What is the probability that a poker hand contains exactly one pair (that is, a poker hand with exactly four different ranks)?

James Chok

Numerade Educator

03:54

Problem 58

What is the probability that a poker hand contains cards of five different ranks but does not contain a flush or a straight?

Dalia Rodriguez

Numerade Educator

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Problem 59

Consider the deck of 40 cards obtained from an ordinary deck of 52 cards by removing the jacks (11s), queens (12s), and kings (13s), where now the 1 (ace) can be used to follow a 10. Compute the probabilities for the various poker hands described in the example in Section $3.6$.

Nick Johnson

Numerade Educator

01:27

Problem 60

A bagel store sells six different kinds of bagels. Suppose you choose 15 bagels at random. What is the probability that your choice contains at least one bagel of each kind? If one of the kinds of bagels is Sesame, what is the probability that your choice contains at least three Sesame bagels?

Ashley Volpe

Numerade Educator

01:34

Problem 61

Consider an 9 -by- 9 board and nine rooks of which five are red and four are blue. Suppose you place the rooks on the board in nonattacking positions at random. What is the probability that the red rooks are in rows $1,3,5,7,9 ?$ What is the probability that the red rooks are both in rows $1,2,3,4,5$ and in columns $1,2,3,4,5 ?$

Amany Waheeb

Numerade Educator

03:35

Problem 62

Suppose a poker hand contains seven cards rather than five. Compute the probabilities of the following poker hands:
(a) a seven-card straight
(b) four cards of one rank and three of a different rank
(c) three cards of one rank and two cards of each of two different ranks
(d) two cards of each of three different ranks, and a card of a fourth rank
(e) three cards of one rank and four cards of each of four different ranks
(f) seven cards each of different rank

Lourence Gonhovi

Numerade Educator

02:58

Problem 63

Four (standard) dice (cubes with $1,2,3,4,5,6$, respectively, dots on their six faces), each of a different color, are tossed, each landing with one of its faces up, thereby showing a number of dots. Determine the following probabilities:
(a) The probability that the total number of dots shown is 6
(b) The probability that at most two of the dice show exactly one dot
(c) The probability that each die shows at least two dots
(d) The probability that the four numbers of dots shown are all different.
(e) The probability that there are exactly two different numbers of dots shown

Lucas Finney

Numerade Educator

01:46

Problem 64

Let $n$ be a positive integer. Suppose we choose a sequence $i_{1}, i_{2}, \ldots, i_{n}$ of integers between 1 and $n$ at random.
(a) What is the probability that the sequence contains exactly $n-2$ different integers?
(b) What is the probability that the sequence contains exactly $n-3$ different integers?

Aman Gupta

Numerade Educator