Painting Exhibit Five Monet paintings are to be displayed

To determine the number of distinct ways two or more experiments can be performed, the principle can be used.

Any ordered arrangement of a given set of objects is called a(n) .

The symbol for n factorial is .

The formula for the number of permutations of n distinct items is n! =

The formula for the number of permutations when r objects are selected from n objects is nPr = .

The number of permutations of n objects, where n1 of the items are identical, n2 of the items are identical, p , nr are identical, is found by .

The notation used to express the number of permutations of five items taken three at a time is .

The notation used to express the number of permutations of seven items taken two at a time is .

In Exercises 920, evaluate the expression. 5!

In Exercises 9-20, evaluate the expression. 6!

In Exercises 9–20, evaluate the expression. \({ }_{4} P_{2}\)

In Exercises 9-20, evaluate the expression. 5P2

In Exercises 920, evaluate the expression. 0!

In Exercises 920, evaluate the expression.7P3

In Exercises 920, evaluate the expression.8P0

In Exercises 920, evaluate the expression.6P0

In Exercises 9–20, evaluate the expression. \({ }_{9} P_{5}\)

In Exercises 9–20, evaluate the expression. \({ }_{3} P_{3}\)

In Exercises 920, evaluate the expression.8P3

In Exercises 920, evaluate the expression.10P6

ATM Codes To use an automated teller machine, you generally must enter a four-digit code, using the digits 09. How many four-digit codes are possible if repetition of digits is permitted?

Routes There are three different routes that Cassidy Cole can walk from home to the post office and two different routes that she can walk from the post office to the bank. How many different routes can Cassidy walk from home to the post office and then to the bank?

Outfits Juan Rivera has three ties, four shirts, and two pairs of pants. How many different outfits can he wear if he chooses one tie, one shirt, and one pair of pants for each outfit?

Passwords Assume that a password to log onto a computer account is to consist of any four digits or letters (repetition is permitted). Determine the number of passwords possible if a) the letters are not case sensitive (that is, a lowercase letter is treated the same as an uppercase letter). b) the letters are case sensitive (that is, an uppercase letter is considered different than the same lowercase letter).

Car Door Locks Some doors on cars can be opened by pressing a correct sequence of buttons. A display of the five buttons by the door handle of a car follows.* 1 - 2 3 - 4 5 - 6 7 - 8 9 - 0 The correct sequence of five buttons must be pressed to unlock the door. a) How many different sequences of five buttons are possible (repetition is permitted)? b) If a sequence of five buttons is pressed at random, determine the probability that the sequence unlocks the door.

Social Security Numbers A social security number consists of nine digits. How many different social security numbers are possible if repetition of digits is permitted?

License Plate In Italy, most car and motorcycle license plates consist of 2 letters, followed by 3 digits, followed by 2 letters. (See photo.) Determine the number of license plates possible if a) repetition is permitted (as it is in Italy). b) repetition is not permitted.

Winning the Trifecta The trifecta at most racetracks consists of selecting the first-, second-, and third-place finishers in a particular race in their proper order. If there are seven entries in the trifecta race, how many tickets must you purchase to guarantee a win?

Choosing Classes Kai Lu plans to enroll in four classes: sociology, chemistry, economics, and humanities. There are seven sociology classes, four chemistry classes, three economics classes, and four humanities classes that fit his schedule. How many different ways can he select his four classes?

Geometric Shapes Consider the five figures shown. In how many different ways can the figures be arranged a) from left to right? b) from top to bottom if placed one under the other? c) from left to right if the triangle is to be placed on the far right? d) from left to right if the circle is to be placed on the far left and the triangle is to be placed on the far right?

Arranging Pictures The six pictures shown are to be placed side by side along a wall. In how many ways can they be arranged from left to right if a) they can be arranged in any order? b) the bird must be on the far left? c) the bird must be on the far left and the giraffe must be next to the bird? d) a four-legged animal must be on the far right?

Club Officers If a club consists of 10 members, how many different arrangements of president, vice-president, and secretary are possible?

Restaurant Positions There are 10 candidates for three positions at Johnny Rockets. One position is for a cook. The second position is for a food server. The third position is for a cashier. If all 10 candidates are equally qualified for the three positions, in how many different ways can the three positions be filled?

ISBN Codes Each book registered in the Library of Congress must have an ISBN code number. For an ISBN number of the form D-DD-DDDDDD-D, where D represents a digit from 0 9, how many different ISBN numbers are possible if repetition of digits is allowed? (See research activity Exercise 72 on page 746.)

Wedding Reception At the reception line of a wedding, the bride, the groom, the best man, the maid of honor, the four ushers, and the four bridesmaids must line up to receive the guests. a) If these individuals can line up in any order, how many arrangements are possible? b) If the groom must be the last in line and the bride must be next to the groom, and the others can line up in any order, how many arrangements are possible? c) If the groom is to be last in line, the bride next to the groom, and males and females are to alternate, how many arrangements are possible?

DJ A disc jockey (DJ) has 9 songs to play. Five are slow songs, and 4 are fast songs. Each song is to be played only once. In how many ways can the DJ play the 9 songs if a) the songs can be played in any order? b) the first song must be a slow song and the last song must be a slow song? c) the first two songs must be fast songs?

Letter Codes In Exercises 37 40, an identification code is to consist of three letters followed by four digits. How many different codes are possible if repetition is not permitted?

Letter Codes In Exercises 37 40, an identification code is to consist of three letters followed by four digits. How many different codes are possible if repetition is permitted?

Letter Codes In Exercises 37 40, an identification code is to consist of three letters followed by four digits. How many different codes are possible if repetition of letters is permitted, repetition of numbers is not permitted, and the first three entries must all be the same letter?

License Plates In Exercises 41 44, a license plate is to consist of three digits followed by two uppercase letters. Determine the number of different license plates possible if repetition of numbers and letters is permitted.

License Plates In Exercises 41 44, a license plate is to consist of three digits followed by two uppercase letters. Determine the number of different license plates possible if repetition of numbers and letters is not permitted.

License Plates In Exercises 41 44, a license plate is to consist of three digits followed by two uppercase letters. Determine the number of different license plates possible if repetition of numbers and letters is not permitted.

License Plates In Exercises 41 44, a license plate is to consist of three digits followed by two uppercase letters. Determine the number of different license plates possible if the first and second digits must be odd, and repetition is not permitted.

License Plate In Italy, most car and motorcycle license plates consist of 2 letters, followed by 3 digits, followed by 2 letters. (See photo.) Determine the number of license plates possible if a) repetition is permitted (as it is in Italy). b) repetition is not permitted.

Possible Phone Numbers A telephone number consists of seven digits with the restriction that the first digit cannot be 0 or 1. a) How many distinct telephone numbers are possible? b) How many distinct telephone numbers are possible with three-digit area codes preceding the seven-digit number, where the first digit of the area code is not 0 or 1? c) With the increasing use of cell phones and paging systems, our society is beginning to run out of usable phone numbers. Various phone companies are developing phone numbers that use 8 digits instead of 7. How many distinct phone numbers can be made using a three digit area code followed by an eight digit phone number, where the first digit of the area code and the phone number cannot be 0 or 1?

Fruit Juice Selection Mrs. Hernandez and her three children go shopping at a local grocery store. Each of the children will be allowed to select one bottle of fruit juice. On the stores shelf there are 12 bottles of fruit juice, and each bottle contains a different type of juice. In how many ways can the selections be made?

Skiing Event A skiing event has 16 participants for the slalom. The 5 participants with the fastest speeds will be listed, in the order of their speed, on the leader board. How many different ways are there for the names to be listed?

History Test In one question on a history test, a student is asked to match 10 dates with 10 events; each date can only be matched with 1 event. In how many different ways can this question be answered?

Color Permutations Determine the number of permutations of the colors in the spectrum that follows. Red Orange Yellow Green Blue Violet Indigo

Drive-Through at a Bank A bank has three drive-through stations. Assuming that each is equally likely to be selected by customers, in how many different ways can the next six drivers select a station?

Computer Systems At a computer store, a customer is considering 5 different computers, 4 different monitors, 7 different printers, and 2 different scanners. Assuming that each of the components is compatible with one another and that one of each is to be selected, determine the number of different computer systems possible.

Selecting Furniture The Johnsons just moved into their new home and are selecting furniture for the family room. They are considering 5 different sofas, 2 different chairs, and 6 different tables. They plan to select one item from each category. Determine the number of different ways they can select the furniture.

Determine the number of permutations of the letters of the word EDUCATION.

Determine the number of permutations of the letters of the word EFFECTIVE.

In how many ways can the letters in the word TENNESSEE be arranged?

In how many ways can the letters in the word COMMUNICATION be arranged?

In how many ways can the digits in the number 4,568,865 be arranged?

In how many ways can the digits of the number 2,142,332 be arranged?

Flag Messages Five different colored flags will be placed on a pole, one beneath another. The arrangement of the colors indicates the message. How many messages are possible if five flags are to be selected from nine different colored flags?

Multiple-Choice Test Keri Kershaw is taking a 12- question multiple-choice exam. Each question has three possible answers, (a), (b), and (c). In how many possible ways can Keri answer the questions?

Batting Order In how many ways can the manager of a National League baseball team arrange his batting order of nine players if a) the pitcher must bat last? b) there are no restrictions?

Painting Exhibit Five Monet paintings are to be displayed in a museum. a) In how many different ways can they be arranged if they must be next to one another? b) In how many different ways can they be displayed if a specific one is to be in the middle?

Car Keys Door keys for a certain automobile are made from a blank key on which five cuts are made. Each cut may be one of five different depths. a) How many different keys can be made? b) If 400,000 of these automobiles are made such that each of the keys determined in part (a) opens the same number of cars, determine how many cars can be opened by a specific key. c) If one of these cars is selected at random, what is the probability that the key selected at random will unlock the door?

Voting On a ballot, each committee member is asked to rank three of eight candidates for recommendation for promotion, giving first, second, and third choices (no ties). What is the minimum number of ballots that must be cast to guarantee that at least two ballots are the same?

Scrabble Nancy Lin, who is playing Scrabble with Dale Grey, has seven different letters. She decides to test each five-letter permutation before her next move. If each permutation takes 5 sec, how long will it take Nancy to check all the permutations?

Scrabble In Exercise 65, assume, of Nancys seven letters, that three are identical and two are identical. How long will it take Nancy to try all different permutations of her seven letters?

Does nPr = nP(n-r) for all whole numbers, where n r ?

Stations There are eight bus stations from town A to town B. How many different single tickets must be printed so that a passenger may purchase a ticket from any station to any other station? A B

Bus Loop How many tickets with different points of origin and destination can be sold on a bus line that travels a loop with 25 stops?

Jumble Many newspapers contain Jumble puzzles, where the letters of a word are given out of order. In Exercises 70 and 71, read the Recreational Math box on page 741 and determine a) the number of possible arrangements of the letters given. b) the word.HEICOC

Jumble Many newspapers contain Jumble puzzles, where the letters of a word are given out of order. In Exercises 70 and 71, read the Recreational Math box on page 741 and determine a) the number of possible arrangements of the letters given. b) the word.ROSEGOC

When a book is published, it is assigned a 10-digit code number called the International Standard Book Number (ISBN). Do research and write a report on how this coding system works.