Chapter 16

Determine the number of possible outcomes. Choosing breakfast of juice, cereal, and fruit from 3 juices, 5 cereals, and 4 fruits

The length of a side of a cube is represented by \((3 x-1) .\) Use the binomial theorem to write a polynomial that represents the volume of the cube.

When adjusted for inflation, the monthly amount that a historic restaurant spent on cleaning for a 30 -year period was normally distributed with a mean of \(\$ 2,100\) and a standard deviation of \(\$ 240 .\) a. What is the probability that the restaurant will spend between \(\$ 2,500\) and \(\$ 2,800\) on cleaning for the month of January? b. If the restaurant has only \(\$ 2,400\) allotted for cleaning for the month of January, what is the probability that it will exceed its budget for cleaning?

There are 3 seniors and 15 juniors in Mrs. Gillis's math class. Three students are chosen at random from the class. a. What is the probability that the group consists of a senior and two juniors? b. If the group consists of a senior and two juniors, what is the probability that Stephanie, a senior, and Jan, a junior, are chosen?

In \(3-22,\) evaluate each expression. $$ _{8} C_{3} \div_{8} C_{5} $$

Determine the number of possible outcomes. Assembling an outfit from 6 shirts, 4 pairs of pants, and 2 pairs of shoes

When \(\$ 100\) are invested at 4\(\%\) interest compounded quarterly, the value of the investment after 3 years is 100\((1+0.01)^{12} .\) Use the binomial theorem to express the value in sigma notation.

Of the 18 students in Mrs. Shusda's math class, 12 take chemistry. If three students are absent from the class today, what is the probability that none of them take chemistry?

In \(3-22,\) evaluate each expression. $$ _{20} C_{5} \div_{20} C_{15} $$

Determine the number of possible outcomes. Choosing a 4-digit entry code using 0–9 if digits cannot be repeated

A machine purchased for \(\$ 75,000\) is expected to decrease in value by 20\(\%\) each year. The value of the machine after \(n\) years is \(75,000(1-0.20)^{n} .\) Use the binomial theorem to express the value of the machine after 5 years in sigma notation.

In 22-24 , use the normal approximation to estimate each probability. Round your answers to three decimal places. A fair die is tossed 70 times. What is the probability of at last 40 even numbers?

In \(23-30,\) find the number of different arrangements that are possible for the letters of each of the following words. FACTOR

Determine the number of possible outcomes. Choosing from 5 flavors of iced tea in 3 different sizes with or without sugar

In 22-24 , use the normal approximation to estimate each probability. Round your answers to three decimal places. The probability that a basketball team will win any game is .5. What is the probability that the team will win at least 12 out of 25 games?

At a carnival, Chrystal is managing a game in which a dart is thrown at a square board with a bull's-eye in the center. The board measures 3 feet by 3 feet and the bull's-eye has a 1 -inch radius. Players who hit the bull's-eye receive a prize. a. Assume that each player is unskilled and each throw is radom but always lands within the square. What is the theoretical probability that a player will hit the bull's-eye? b. At the end of the day, Chrystal finds that she gave out 72 prizes for \(1,270\) throws. What is the experimental probability that a player hit the bull's-eye? c. Are the theoretical probability and the empirical probability the same? If not, explain why they are different.

In \(23-30,\) find the number of different arrangements that are possible for the letters of each of the following words. DIVIDE

Determine the number of possible outcomes. Ordering an ice cream cone from a choice of 31 flavors, 3 types of cone, with or without one of 4 toppings

An irregular figure is drawn on a graph within a square that measures 6 inches on each side. The theoretical probability that the coordinates of a point that lies within the square also lies within the irregular figure is \(\frac{4}{9}\) . What is the area of the irregular figure?

In \(23-30,\) find the number of different arrangements that are possible for the letters of each of the following words. EXCEED

Determine the number of possible outcomes. Making a 7-character license plate using the letters of the alphabet and the digits 1–4 if the first three characters must be non-repeating letters and the remaining four are digits that may repeat

A group of friends are playing a game in which each player chooses a number from 1 to 6 and takes a turn tossing a die until the chosen number appears on the die. The required number of tosses is the person's score for that round of play. a. What is the probability that a player gets a score of 3 on the first round? b. What is the probability that a player gets a score of 5 on the first round? c. What is the probability of a score of \(n\) ? d. Show that the probabilities of scores that are consecutive integers from 1 to 5 form a geometric sequence.

In \(23-30,\) find the number of different arrangements that are possible for the letters of each of the following words. ABSCISSA

Determine the number of possible outcomes. Seating Andy, Brenda, Carlos, Dabeed, and Eileen in a row of 5 seats

A carton of 24 eggs contains 4 eggs with double yolks. If 3 eggs are selected at random, determine each probability to the nearest hundredth: a. All 3 eggs will have single yolks. b. 2 eggs will have single yolks and 1 egg will have a double yolk.

In \(23-30,\) find the number of different arrangements that are possible for the letters of each of the following words. RANDOMLY

Determine the number of possible outcomes. Seating Andy, Brenda, Carlos, Dabeed, and Eileen in a row of 5 seats with Andy and Eileen occupying end seats

In \(23-30,\) find the number of different arrangements that are possible for the letters of each of the following words. REMEMBERED

In how many ways can 5 new bus passengers choose seats if there are 8 empty seats?

In \(23-30,\) find the number of different arrangements that are possible for the letters of each of the following words. STATISTICS

In how many ways can 3 different job openings be filled if there are 8 applicants?

In \(23-30,\) find the number of different arrangements that are possible for the letters of each of the following words. MATHEMATICS

In how many ways can 5 roles in the school play be filled if there are 9 possible people trying out? (Each role is to be played by a different person.)

A box contains 9 red, 4 blue, and 6 vellow chips. In how many ways can 6 chips be chosen if: a. all 6 chips are red? c. 2 chips are blue? e. 4 chips are yellow? g. 3 chips are red and 3 chips are blue? b. all 6 chips are yellow? d. 3 chips are red? f. there are 2 chips of each color? h. 5 chips are red and 1 chip is yellow?

From the set {2, 3, 4, 5, 6}, a number is drawn and replaced. Then a second number is drawn. How many two-digit numbers can be formed?

In \(32-37,\) determine the number of different arrangements. The finishing order of 7 runners in a race

In a class of 22 students, the teacher calls on a student to give the answer to the first homework problem and then calls on a student to give the answer to the second homework problem. a. How many possible choices could the teacher have made if the same student was not called on twice? b. How many possible choices could the teacher have made if the same student may have been called on twice?

In \(32-37,\) determine the number of different arrangements. Seating of 5 students in a row of 9 chairs

A manufacturer produces jeans in 9 sizes, 7 different shades of blue, and 6 different leg widths. If a branch store manager orders two pairs of each possible type, how many pairs of jeans will be in stock?

In \(32-37,\) determine the number of different arrangements. Stacking 6 red, 4 yellow, and 2 blue t-shirts

In \(32-37,\) determine the number of different arrangements. The order in which a student answers 8 out of 10 test questions

On a certain ski run, there are 8 places where a skier can choose to go to the left or to the right. In how many different ways can the skier cover the run?

In \(32-37,\) determine the number of different arrangements. Making a row of coins using 4 pennies, 3 nickels, and 3 dimes

Stan has letter tiles AMTE. a. How many different ways can Stan arrange all four tiles? b. How many different arrangements begin with a vowel?

In \(32-37,\) determine the number of different arrangements. Rating 6 employees in order of their friendliness

Six students will be seated in a row in the classroom. a. How many different ways can they be seated? b. If one student forgot his eyeglasses and must occupy the front seat, how many different seatings are possible?

To duplicate a key, a locksmith begins with a dummy key that has several sections. The locksmith grinds a specific pattern into each section. a. A particular brand of house key includes 6 sections, and there are 4 possible patterns for each section. How many different house keys are possible? b. A desk key has 3 sections, and 64 different keys are possible. How many patterns are available for each section if each section has the same number of possible patterns?

\(\ln 38-43,\) solve for \(x\) $$ _{x} P_{6}=30\left(_{x} P_{4}\right) $$

A physicist, a chemist, a biologist, an astronomer, a geologist, and a mathematician are guest speakers at a government-sponsored forum on scientific research in the twenty-first century. The speakers will be seated in a row on a raised platform at the front of the meeting room. a. How many different ways can the speakers be seated? b. The astronomer and the mathematician have co-written a paper that they will present. How many ways can the speakers be seated if the astronomer and the mathematician wish to sit side-by-side?

\(\ln 38-43,\) solve for \(x\) $$ _{13} P_{5}=1,287\left(_{x} P_{x}\right) $$