Algebra and Trigonometry

James Stewart, Lothar Redlin, Saleem Watson

Chapter 14 Counting and Probability - all with Video Answers

Educators

Section 1

Counting Principles

Problem 1

The Fundamental Counting Principle says that if one event can occur in $m$ ways and a second event can occur in $n$ ways, then the two events can occur in order in _____ x _____ ways. So if you have two choices for shoes and three choices for hats, then the number of different shoe-hat combinations you can wear is _____ x ____ = _____

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01:46

Problem 2

The Fundamental Counting Principle also applies to three or more events in order. So if you have 2 choices for shoes, 5 choices for pants, 4 choices for shirts, and 3 choices for hats, then the number of different shoe-pants-shirt-hat outfits you can wear is ______x ______ x______x______ = ______

Maggie H.

Numerade Educator

03:48

Problem 3

Ice-Cream Cones A vendor sells ice cream from a cart on the boardwalk. He offers vanilla, chocolate, strawberry, and pistachio ice cream, served in either a waffe, sugar, or plain cone. How many different single-scoop ice-cream cones can you buy from this vendor?

Scott A.

Numerade Educator

01:46

Problem 4

Three-Letter Words How many three-letter "words" (strings of letters) can be formed by using the 26 letters of the alphabet if repetition of letters
$$\begin{array}{lll}{\text { (a) is allowed? }} & {\text { (b) is not allowed? }}\end{array}$$

Kyler G.

Numerade Educator

01:42

Problem 5

Three-Letter Words How many three-letter "words" (strings of letters) can be formed by using the letters $W X Y Z$ if repetition of letters
$$\begin{array}{lll}{\text { (a) is allowed? }} & {\text { (b) is not allowed? }}\end{array}$$

Savannah L.

Numerade Educator

01:30

Problem 6

Horse Race Eight horses are entered in a race.
(a) How many different orders are possible for completing the race?
(b) In how many different ways can first, second, and third places be decided? (Assume that there is no tie.)

Mumina S.

Numerade Educator

00:54

Problem 7

Multiple-Choice Test A multiple-choice test has five questions with four choices for each question. In how many different ways can the test be completed?

Kyler G.

Numerade Educator

01:07

Problem 8

Phone Numbers Telephone numbers consist of seven digits; the first digit cannot be 0 or 1. How many telephone numbers are possible?

Kyler G.

Numerade Educator

02:48

Problem 9

Running a Race In how many different ways can a race with five runners be completed? (Assume that there is no tie.)

Shenade G.

Numerade Educator

Problem 10

Seating Order In how many ways can five people be seated in a row of five seats?

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02:26

Problem 11

Restaurant Meals A restaurant offers the items listed in the table. How many different meals consisting of a main course, a drink, and a dessert can be selected at this restaurant?
table can't copy

Destin P.

Numerade Educator

00:56

Problem 12

Lining Up Books In how many ways can five different mathematics books be placed next to each other on a shelf?

Kyler G.

Numerade Educator

Problem 13

Multiple Routes Towns $\mathrm{A}, \mathrm{B}, \mathrm{C},$ and $\mathrm{D}$ are located in such a way that there are four roads from $A$ to $B$ , five roads from $B$ to $C,$ and six roads from $C$ to D. How many routes are there from town $A$ to town D via towns $B$ and $C ?$

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03:04

Problem 14

Birth Order In a family of four children, how many different boy-girl birth-order combinations are possible? (The birth orders $B B B G$ and $B B G B$ are different.)

Christy G.

Numerade Educator

03:25

Problem 15

Flipping a Coin $A$ coin is flipped five times, and the resulting sequence of heads and tails is recorded. How many such sequences are possible?

Cora C.

Numerade Educator

00:50

Problem 16

Rolling a Pair of Dice A red die and a white die are rolled, and the numbers that show are recorded. How many different out-comes are possible? (The singular form of the word dice is die.)

Faizaan S.

Numerade Educator

00:34

Problem 17

Rolling Three Dice A red die, a blue die, and a white die are rolled, and the numbers that show are recorded. How many different outcomes are possible?

Kyler G.

Numerade Educator

03:20

Problem 18

Picking Cards Two cards are chosen in order from a deck. In how many ways can this be done if
(a) the first card must be a spade and the second must be a heart?
(b) both cards must be spades?

Justin J.

Numerade Educator

00:40

Problem 19

Choosing Outfits A girl has 5 skirts, 8 blouses, and 12 pairs of shoes. How many different skirt-blouse-shoes outfits can she wear? (Assume that each item matches all the others, so she is willing to wear any combination.)

Kyler G.

Numerade Educator

01:34

Problem 20

ID Numbers A company's employee ID number system consists of one letter followed by three digits. How many different ID numbers are possible with this system?

Justin S.

Numerade Educator

00:57

Problem 21

ID Numbers A company has 2844 employees. Each employee is to be given an ID number that consists of one letter followed by two digits. Is it possible to give each employee a different ID number using this scheme? Explain.

Charles C.

Numerade Educator

Problem 22

Pitchers and Catchers An all-star baseball team has a roster of seven pitchers and three catchers. How many pitcher-catcher pairs can the manager select from this roster?

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01:05

Problem 23

License Plates Standard automobile license plates in California display a nonzero digit, followed by three letters, followed by three digits. How many different standard plates are possible in this system?

Kyler G.

Numerade Educator

00:42

Problem 24

Combination Lock A combination lock has 60 different positions. To open the lock, the dial is turned to a certain number in the clockwise direction, then to a number in the counterclockwise direction, and finally to a third number in the clockwise direction. If successive numbers in the combination cannot be the same, how many different combinations are possible?

Christy G.

Numerade Educator

Problem 25

True-False Test $A$ true-false test contains ten questions. In how many different ways can this test be completed?

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Problem 26

Ordering a Car An automobile dealer offers five models. Each model comes in a choice of four colors, three types of stereo equipment, with or without air conditioning, and with or without a sunroof. In how many different ways can a customer order an auto from this dealer?

Jasmine H.

Numerade Educator

Problem 27

Classifications The registrar at a certain university classifies students according to a major, minor, year $(1,2,3,4),$ and sex $(\mathrm{M}, \mathrm{F}) .$ Each student must choose one major and either one or no minor from the 32 fields taught at this university. How many different student classifications are possible?

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Problem 28

Monograms How many monograms consisting of three initials are possible?

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01:03

Problem 29

License Plates A state has registered 8 million automobiles. To simplify the license plate system, a state employee suggests that each plate display only two letters followed by three digits. Will this system create enough different license plates for all the vehicles that are registered?

Kyler G.

Numerade Educator

01:58

Problem 30

License Plates A state license plate design has six places. Each plate begins with a fixed number of letters, and the remaining places are filled with digits. (For example, one letter followed by five digits, two letters followed by four digits, and so on.) The state has 17 million registered vehicles.
(a) The state decides to change to a system consisting of one letter followed by five digits. Will this design allow for enough different plates to accommodate all the vehicles that are registered?
(b) Find a system that will be sufficient if the smallest possible number of letters is to be used.

Kyler G.

Numerade Educator

00:45

Problem 31

Class Executive In how many ways can a president, vice president, and secretary be chosen from a class of 30 students?

Kyler G.

Numerade Educator

Problem 32

Class Executive In how many ways can a president, vice president, and secretary be chosen from a class of 20 females and 30 males if the president must be a female and the vice president must be a male?

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