Precalculus

McGraw-Hill Ryerson

Chapter 11

Permutations, Combinations, and the Binomial Theorem - all with Video Answers

Educators

Section 1

Permutations

Problem 1

Use an organized list or a tree diagram to identify the possible arrangements for
a) the ways that three friends, Jo, Amy, and Mike, can arrange themselves in a row.
b) the ways that you can arrange the digits
$2,5,8,$ and 9 to form two-digit numbers.
c) the ways that a customer can choose a starter, a main course, and a dessert from the following menu.

Arushi Sahay

Arushi Sahay

Numerade Educator

Problem 2

Evaluate each expression.
a) $_{a} P_{2}$
b) $P_{5}$
c) $_{6} P_{6}$
d) $_{4} P_{1}$

Arushi Sahay

Numerade Educator

Problem 3

Show that $4 !+3 ! \neq(4+3) !$

Arushi Sahay

Numerade Educator

Problem 4

What is the value of each expression?
a) $9 !$
b) $\frac{9 !}{5 ! 4 !}$
c) $(5 !)(3 !)$
d) $6(4 !)$
e) $\frac{102 !}{100 ! 2 !}$
f) $7 !-5 !$

Arushi Sahay

Numerade Educator

Problem 5

In how many different ways can you arrange all of the letters of each word?
a) hoodie
b) decided
c) aqilluqqaaq
d) deeded
e) puppy
f) baguette

Arushi Sahay

Numerade Educator

Problem 6

Four students are running in an election for class representative on the student council. In how many different ways can the four names be listed on the ballot?????? hemes bed on the balli????

Arushi Sahay

Numerade Educator

Problem 7

Solve for the variable.
a) $_{n} P_{2}=30$
b) $_{n} P_{3}=990$
c) $_{6} P_{r}=30$
d) $2\left(p_{n}\right)=60$

Umar Sohail Qureshi

Umar Sohail Qureshi

Numerade Educator

Problem 8

Determine the number of pathways from A to B.
a) Move only down or to the right.
b) Move only up or to the right.
c) Move only up or to the left.

Arushi Sahay

Numerade Educator

Problem 9

Describe the cases you could use to solve each problem. Do not solve.
a) How many 3 -digit even numbers greater than 200 can you make using the digits
$1,2,3,4,$ and $5 ?$
b) How many four-letter arrangements beginning with either B or E and ending with a vowel can you make using the letters $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{E}, \mathrm{U},$ and $\mathrm{G} ?$

Arushi Sahay

Numerade Educator

Problem 10

In how many ways can four girls and two boys be arranged in a row if
a) the boys are on each end of the row?
b) the boys must be together?
c) the boys must be together in the middle of the row?

Arushi Sahay

Numerade Educator

Problem 11

In how many ways can seven books be arranged on a shelf if
a) the books are all different?
b) two of the books are identical?
c) the books are different and the mathematics book must be on an end?
d) the books are different and four particular books must be together?

Arushi Sahay

Numerade Educator

Problem 12

How many six-letter arrangements can you make using all of the letters $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}$
and $\mathrm{F},$ without repetition? Of these, how many begin and end with a consonant?

Arushi Sahay

Numerade Educator

Problem 13

A national organization plans to issue its members a 4 -character ID code. The first character can be any letter other than O. The last 3 characters are to be 3 different digits. If the organization has 25300 members, will they be able to assign each member a different ID code? Explain.

Arushi Sahay

Numerade Educator

Problem 14

Iblauk lives in Baker Lake, Nunavut. She makes oven mitts to sell. She has wool duffel in red, dark blue, green, light blue, and yellow for the body of each mitt. She has material for the wrist edge in dark green, pink, royal blue, and red. How many different colour.

Arushi Sahay

Numerade Educator

Problem 15

You have forgotten the number sequence to your lock. You know that the correct code is made up of three numbers (right-left-right). The numbers can be from 0 to 39 and repetitions are allowed. If you can test one number sequence every $15 \mathrm{s},$ how long will it take to test all possible number sequences? Express your answer in hours.

Arushi Sahay

Numerade Educator

Problem 16

Jodi is parking seven different types of vehicles side by side facing the display window at the dealership where she works.
a) In how many ways can she park the vehicles?
b) In how many ways can she park them so that the pickup truck is next to the hybrid car?
c) In how many ways can she park them
so that the convertible is not next to the subcompact?

Arushi Sahay

Numerade Educator

Problem 17

a) How many arrangements using all of the letters of the word parallel are possible?
b) How many of these arrangements have all of the $I$ s together?

Arushi Sahay

Numerade Educator

Problem 18

The number of different permutations using all of the letters in a particular set is given by $\frac{5 !}{2 ! 2 !}$
a) Create a set of letters for which this is true.
b) What English word could have this number of arrangements of its letters?

Arushi Sahay

Numerade Educator

Problem 19

How many integers from 3000 to 8999, inclusive, contain no $7 \mathrm{s} ?$

Arushi Sahay

Numerade Educator

Problem 20

Postal codes in Canada consist of three letters and three digits. Letters and digits alternate, as in the code R7B 5K1.
a) How many different postal codes are possible with this format?
b) Do you think Canada will run out of postal codes? Why or why not?

Arushi Sahay

Numerade Educator

Problem 21

Cent mille milliards de poèmes (One Hundred Million Million Poems) was written in 1961 by Raymond Queneau, a French poet, novelist, and publisher. The book is 10 pages long, with 1 sonnet per page. A sonnet is a poem with 14 lines. Each line of every sonnet can be replaced by a line at the same position on a different page. Regardless of which lines are used, the poem makes sense.
a) How many arrangements of the lines are possible for one sonnet?
b) Is the title of the book of poems reasonable? Explain.

Sheryl Ezze

Numerade Educator

Problem 22

Use your understanding of factorial notation and the symbol $_{n} P_{r}$ to solve each equation.
a) $_{3} P_{r}=3 !$
b) $_{z_{r}}=7 !$
c) $_{n} P_{3}=4\left(_{n-1} P_{2}\right)$
d) $n\left(_{5} P_{3}\right)=_{7} P_{5}$

Arushi Sahay

Numerade Educator

Problem 23

Use $_{n} P_{n}$ to show that $0 !=1$

Arushi Sahay

Numerade Educator

Problem 24

Explain why $_{3} P_{5}$ gives an error message when evaluated on a calculator.

Arushi Sahay

Numerade Educator

Problem 25

How many odd numbers of at most three digits can be formed using the digits $0,1,2,3,4,$ and 5 without repetitions?

Aman Gupta

Numerade Educator

Problem 26

How many even numbers of at least four digits can be formed using the digits $0,1,2,3,$ and 5 without repetitions?

Arushi Sahay

Numerade Educator

Problem 27

How many integers between 1 and 1000 do not contain repeated digits?

Arushi Sahay

Numerade Educator

Problem 28

A box with a lid has inside dimensions of $3 \mathrm{cm}$ by $2 \mathrm{cm}$ by $1 \mathrm{cm} .$ You have four identical blue cubes and two identical yellow cubes, each $1 \mathrm{cm}$ by $1 \mathrm{cm}$ by $1 \mathrm{cm} .$ How many different six-cube arrangements of blue and yellow cubes are possible? You must be able to close the lid after any arrangement. The diagram below shows one possible arrangement. Show two different ways to solve the problem.

Matthew Biollo

Numerade Educator

Problem 29

You have two colours of paint. In how many different ways can you paint the faces of a cube if each face is painted? Painted cubes are considered to be the same if you can rotate one cube so that it matches the other one exactly.

Taylor Jordan

Numerade Educator

Problem 30

Nine students take a walk on four consecutive days. They always walk in rows of three across. Show how to arrange the students so that each student walks only once in a row with any two other students during the four-day time frame. In other words, no three-across triplets are repeated.

Arushi Sahay

Numerade Educator

Problem 31

If $100 !$ is evaluated, how many zeros are at the end of the number? Explain how you know.

Pagadala Kishore Reddy

Pagadala Kishore Reddy

Numerade Educator

Problem 32

There are five people: $A, B, C, D,$ and $E$
The following pairs know each other: $\mathrm{A}$ and $\mathrm{C}, \mathrm{B}$ and $\mathrm{C}, \mathrm{A}$ and $\mathrm{D}, \mathrm{D}$ and $\mathrm{E},$ and C and D.
a) Arrange the five people in a row so that nobody is next to a stranger.
b) How many different arrangements are possible such that nobody is next to a stranger?
c) The five people are joined by a sixth person, $F$, who knows only A. In how many ways can the six people stand in a row if nobody can be next to a stranger? Explain your answer.

Arushi Sahay

Numerade Educator