Problem 13

How many different types of homes are available if a builder offers a choice of 6 basic plans, 3 roof styles, and 2 exterior finishes?

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Problem 14

A menu offers a choice of 3 salads, 8 main dishes, and 7 desserts. How many different meals consisting of one salad, one main dish, and one dessert are possible?

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Problem 15

A couple has narrowed down the choice of a name for their new baby to 4 first names and 5 middle names. How many different first- and middle-name arrangements are possible?

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Problem 16

In a club with 16 members, how many ways can a slate of 3 officers consisting of president, vice-president, and secretary/ treasurer be chosen?

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Problem 18

Explain the difference between distinguishable and indistinguishable permutations.

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Problem 19

In Example 6, there are six 3-letter permutations of the letters A, B, and C. How many 3-letter subsets (unordered groups of letters) are there?

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Problem 20

In Example 6, how many unordered 2-letter subsets of the letters A, B, and C are there?

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Problem 21

Find the number of distinguishable permutations of the letters in each word.

a. initial b. little c. decreed

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Problem 22

A printer has $5 \mathrm{As}, 4 \mathrm{B}^{\prime} \mathrm{s}, 2 \mathrm{C}^{\prime} \mathrm{s},$ and 2 $\mathrm{D}$ 's. How many differ- ent "words" are possible that use all these letters? (A "word"

does not have to have any meaning here.)

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Problem 23

Wing has different books to arrange on a shelf: 4 blue, 3 green, and 2 red.

a. In how many ways can the books be arranged on a shelf?

b. If books of the same color are to be grouped together, how many arrangements are possible?

c. In how many distinguishable ways can the books be arranged if books of the same color are identical but need not be grouped together?

d. In how many ways can you select 3 books, one of each color, if the order in which the books are selected does not matter?

e. In how many ways can you select 3 books, one of each color, if the order in which the books are selected matters?

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Problem 24

A child has a set of differently shaped plastic objects. There are 3 pyramids, 4 cubes, and 7 spheres.

a. In how many ways can she arrange the objects in a row if each is a different color?

b. How many arrangements are possible if objects of the same shape must be grouped together and each object is a different color?

c. In how many distinguishable ways can the objects be arranged in a row if objects of the same shape are also the same color but need not be grouped together?

d. In how many ways can you select 3 objects, one of each shape, if the order in which the objects are selected does not matter and each object is a different color?

e. In how many ways can you select 3 objects, one of each shape, if the order in which the objects are selected matters and each object is a different color?

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Problem 25

If you already knew the value of how could you find the value of quickly?

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Problem 26

Given that $450 !$ is approximately equal to $1.7333687 \times 10^{1000}$ (to 8 digits of accuracy), find $451 !$ to 7 digits of accuracy.

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Problem 27

When calculating the number of ending zeros in the answer can be determined prior to calculating the actual number by finding the number of times 5 can be factored from For example, only has one 5 occurring in its calculation, and so there is only one ending zero in 5040. The number has two 5’s (one from the 5 and one from the 10) and so there must be two ending zeros in the answer 3,628,800. Use this idea to determine the number of zeros that occur in the following factorials, and then explain why this works.

$\begin{array}{llll}{\text { a. } 13 !} & {\text { b. } 27 !} & {\text { c. } 75 !}\end{array}$

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Problem 28

Because of the view screen, calculators only show a fixed number of digits, often 10 digits. Thus, an approximation of a number will be shown by only including the 10 largest place values of the number. Using the ideas from the previous exercise, determine if the following numbers are correct or if they are incorrect by checking if they have the correct number of ending zeros. (Note: Just because a number has the correct number of zeros does not imply that it is correct.)

a. $12 !=479,001,610$

b. $23 !=25,852,016,740,000,000,000,000$

c. $15 !=1,307,643,680,000$

d. $14 !=87,178,291,200$

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Problem 29

Some students find it puzzling that $0 !=1,$ and think that $0 !$ should equal $0 .$ If this were true, what would be the value of $P(4,4)$ using the permutations formula?

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Problem 30

Automobile Manufacturing An automobile manufacturer produces 8 models, each available in 7 different exterior colors, with 4 different upholstery fabrics and 5 interior colors. How many varieties of automobile are available?

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Problem 31

Marketing In a recent marketing campaign, Olive Garden Italian Restaurant® offered a “Never-Ending Pasta Bowl.” The customer could order an array of pasta dishes, selecting from 7 types of pasta and 6 types of sauce, including 2 with meat.

a. If the customer selects one pasta type and one sauce type, how many different “pasta bowls” can a customer order?

b. How many different “pasta bowls” can a customer order without meat?

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Problem 32

Investments Natalie Graham’s financial advisor has given her a list of 9 potential investments and has asked her to select and rank her favorite five. In how many different ways can she do this?

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Problem 33

Scheduling A local television station has eleven slots for commercials during a special broadcast. Six restaurants and 5 stores have bought slots for the broadcast.

a. In how many ways can the commercials be arranged?

b. In how many ways can the commercials be arranged so that the restaurants are grouped together and the stores are grouped together?

c. In how many ways can the commercials be arranged so that the restaurant and store commercials are alternating?

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Problem 34

Drug Sequencing Twelve drugs have been found to be effective in the treatment of a disease. It is believed that the sequence in which the drugs are administered is important in the effectiveness of the treatment. In how many different sequences can 5 of the 12 drugs be administered?

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Problem 35

Insect Classification A biologist is attempting to classify 52,000 species of insects by assigning 3 initials to each species. Is it possible to classify all the species in this way? If not, how many initials should be used?

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Problem 36

Science Conference At an annual college science conference, student presentations are scheduled one after another in the afternoon session. This year, 5 students are presenting in biology, 5 students are presenting in chemistry, and 2 students are presenting in physics.

a. In how many ways can the presentations be scheduled?

b. In how many ways can the presentations be scheduled so that each subject is grouped together?

c. In how many ways can the presentations be scheduled if the conference must begin and end with a physics presentation?

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Problem 37

Social Science Experiment In an experiment on social inter- action, 6 people will sit in 6 seats in a row. In how many ways can this be done?

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Problem 38

Election Ballots In an election with 3 candidates for one office and 6 candidates for another office, how many different ballots may be printed?

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Problem 39

Baseball Teams A baseball team has 19 players. How many 9-player batting orders are possible?

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Problem 40

Union Elections A chapter of union Local 715 has 35 members. In how many different ways can the chapter select a president, a vice-president, a treasurer, and a secretary?

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Problem 41

Programming Music A concert to raise money for an economics prize is to consist of 5 works: 2 overtures, 2 sonatas, and a piano concerto.

a. In how many ways can the program be arranged?

b. In how many ways can the program be arranged if an overture must come first?

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Problem 42

Programming Music A zydeco band from Louisiana will play 5 traditional and 3 original Cajun compositions at a concert. In how many ways can they arrange the program if

a. they begin with a traditional piece?

b. an original piece will be played last?

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Problem 43

Radio Station Call Letters How many different 4-letter radio station call letters can be made if

a. the first letter must be K or W and no letter may be repeated?

b. repeats are allowed, but the first letter is K or W?

c. the first letter is K or W, there are no repeats, and the last letter is R?

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Problem 44

Telephone Numbers How many 7-digit telephone numbers are possible if the first digit cannot be zero and

a. only odd digits may be used?

b. the telephone number must be a multiple of 10 (that is, it must end in zero)?

c. the telephone number must be a multiple of 100?

d. the first 3 digits are 481?

e. no repetitions are allowed?

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Problem 45

Telephone Area Codes Several years ago, the United States began running out of telephone numbers. Telephone companies introduced new area codes as numbers were used up, and eventually almost all area codes were used up.

a. Until recently, all area codes had a 0 or 1 as the middle digit, and the first digit could not be 0 or 1. How many area codes are there with this arrangement? How many telephone numbers does the current 7-digit sequence permit per area code? (The 3-digit sequence that follows the area

code cannot start with 0 or 1. Assume there are no other restrictions.)

b. The actual number of area codes under the previous system was 152. Explain the discrepancy between this number and your answer to part a.

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Problem 46

The shortage of area codes was avoided by removing the restriction on the second digit. (This resulted in problems for some older equipment, which used the second digit to determine that a long-distance call was being made.) How many area codes are available under the new system?

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Problem 47

License Plates For many years, the state of California used 3 letters followed by 3 digits on its automobile license plates.

a. How many different license plates are possible with this arrangement?

b. When the state ran out of new numbers, the order was reversed to 3 digits followed by 3 letters. How many new license plate numbers were then possible?

c. Several years ago, the numbers described in b were also used up. The state then issued plates with 1 letter followed by 3 digits and then 3 letters. How many new license plate numbers will this provide?

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Problem 48

Social Security Numbers A social security number has 9 digits. How many social security numbers are there? The U.S. population in 2010 was about 309 million. Is it possible for every U.S. resident to have a unique social security number? (Assume no restrictions.)

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Problem 49

Postal Zip Codes The U.S. Postal Service currently uses 5-digit zip codes in most areas. How many zip codes are possible if there are no restrictions on the digits used? How many would be possible if the first number could not be 0?

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Problem 50

Postal Zip Codes The U.S. Postal Service is encouraging the use of 9-digit zip codes in some areas, adding 4 digits after the usual 5-digit code. How many such zip codes are possible with no restrictions?

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Problem 51

Games The game of Sets uses a special deck of cards. Each card has either one, two, or three identical shapes, all of the same color and style. There are three possible shapes: squiggle, diamond, and oval. There are three possible colors: green, purple, and red. There are three possible styles: solid, shaded,

or outline. The deck consists of all possible combinations of shape, color, style, and number of shapes. How many cards are in the deck? Source: Sets.

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Problem 52

Games In the game of Scattergories, the players take 12 turns. In each turn, a 20-sided die is rolled; each side has a letter. The players must then fill in 12 categories (e.g., vegetable, city, etc.) with a word beginning with the letter rolled. Considering that a game consists of 12 rolls of the 20-sided die, and that rolling the same side more than once is allowed, how many possible games are there? Source: Milton Bradley.

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Problem 53

Games The game of Twenty Questions consists of asking 20 questions to determine a person, place, or thing that the other person is thinking of. The first question, which is always “Is it an animal, vegetable, or mineral?” has three possible answers. All the other questions must be answered “Yes” or “No.” How many possible objects can be distinguished in this game, assuming that all 20 questions are asked? Are 20 questions enough?

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Problem 54

Traveling Salesman In the famous Traveling Salesman Problem, a salesman starts in any one of a set of cities, visits every city in the set once, and returns to the starting city. He would like to complete this circuit with the shortest possible distance.

a. Suppose the salesman has 10 cities to visit. Given that it does not matter what city he starts in, how many different circuits can he take?

b. The salesman decides to check all the different paths in part a to see which is shortest, but realizes that a circuit has the same distance whichever direction it is traveled. How many different circuits must he check?

c. Suppose the salesman has 70 cities to visit. Would it be feasible to have a computer check all the different circuits?

Explain your reasoning.

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Problem 55

Circular Permutations Circular permutations arise in applications involving arrangements around a closed loop, as in the previous exercise. Here are two examples.

a. A ferris wheel has 20 seats. How many ways can 20 students arrange themselves on the ferris wheel if each student takes a different seat? We consider two arrangements to be identical if they differ only by rotations of the wheel.

b. A necklace is to be strung with 15 beads, each of a different color. In how many ways can the beads be arranged? We consider two arrangements to be identical if they differ only by rotations of the necklace or by flipping the necklace over. (Hint: If every arrangement is counted twice, the correct number of arrangements can be found by dividing by 2.)

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