License Plate Numbers In a certain state, each automobile license plate number consists

Fill in the blanks. The ________ ________ ________ states that if there are ways for one event to occur and ways for a second event to occur, there are ways for both events to occur.

Fill in the blanks. An ordering of elements is called a ________ of the elements.

Fill in the blanks. The number of permutations of elements taken at a time is given by the formula ________.

Fill in the blanks. The number of ________ ________ of objects is given by

Fill in the blanks. When selecting subsets of a larger set in which order is not important, you are finding the number of ________ of elements taken at a time.

In Exercises 18, determine the number of ways a computer can randomly generate one or more such integers from 1 through 12. An integer that is divisible by 3

In Exercises 18, determine the number of ways a computer can randomly generate one or more such integers from 1 through 12. Two distinct integers whose sum is 9

In Exercises 18, determine the number of ways a computer can randomly generate one or more such integers from 1 through 12. Two distinct integers whose sum is 8

Entertainment Systems A customer can choose one of three amplifiers, one of two compact disc players, and one of five speaker models for an entertainment system. Determine the number of possible system configurations.

Job Applicants A college needs two additional faculty members: a chemist and a statistician. In how many ways can these positions be filled if there are five applicants for the chemistry position and three applicants for the statistics position?

Course Schedule A college student is preparing a course schedule for the next semester. The student may select one of two mathematics courses, one of three science courses, and one of five courses from the social sciences and humanities. How many schedules are possible?

Aircraft Boarding Eight people are boarding an aircraft. Two have tickets for first class and board before those in the economy class. In how many ways can the eight people board the aircraft?

True-False Exam In how many ways can a six-question true-false exam be answered? (Assume that no questions are omitted.)

True-False Exam In how many ways can a 12-question true-false exam be answered? (Assume that no questions are omitted.)

License Plate Numbers In the state of Pennsylvania, each standard automobile license plate number consists of three letters followed by a four-digit number. How many distinct license plate numbers can be formed in Pennsylvania?

License Plate Numbers In a certain state, each automobile license plate number consists of two letters followed by a four-digit number. To avoid confusion between O and zero and between I and one, the letters O and I are not used. How many distinct license plate numbers can be formed in this state?

Three-Digit Numbers How many three-digit numbers can be formed under each condition? (a) The leading digit cannot be zero. (b) The leading digit cannot be zero and no repetition of digits is allowed. (c) The leading digit cannot be zero and the number must be a multiple of 5. (d) The number is at least 400.

Four-Digit Numbers How many four-digit numbers can be formed under each condition? (a) The leading digit cannot be zero. (b) The leading digit cannot be zero and no repetition of digits is allowed. (c) The leading digit cannot be zero and the number must be less than 5000. (d) The leading digit cannot be zero and the number must be even.

Combination Lock A combination lock will open when the right choice of three numbers (from 1 to 40, inclusive) is selected. How many different lock combinations are possible?

Combination Lock A combination lock will open when the right choice of three numbers (from 1 to 50, inclusive) is selected. How many different lock combinations are possible?

Concert Seats Four couples have reserved seats in a row for a concert. In how many different ways can they be seated if (a) there are no seating restrictions? (b) the two members of each couple wish to sit together?

Single File In how many orders can four girls and four boys walk through a doorway single file if (a) there are no restrictions? (b) the girls walk through before the boys?

In Exercises 2328, evaluate

In Exercises 2328, evaluate

In Exercises 2328, evaluate

In Exercises 2328, evaluate

In Exercises 2328, evaluate

In Exercises 2328, evaluate

In Exercises 29 and 30, solve for n

In Exercises 29 and 30, solve for n

In Exercises 3136, evaluate using a graphing utility.

In Exercises 3136, evaluate using a graphing utility.

In Exercises 3136, evaluate using a graphing utility.

In Exercises 3136, evaluate using a graphing utility.

In Exercises 3136, evaluate using a graphing utility.

In Exercises 3136, evaluate using a graphing utility.

Posing for a Photograph In how many ways can five children posing for a photograph line up in a row?

Riding in a Car In how many ways can six people sit in a six-passenger car?

Choosing Officers From a pool of 12 candidates, the offices of president, vice-president, secretary, and treasurer will be filled. In how many different ways can the offices be filled?

Assembly Line Production There are four processes involved in assembling a product, and these processes can be performed in any order. The management wants to test each order to determine which is the least time-consuming. How many different orders will have to be tested?

In Exercises 4144, find the number of distinguishable permutations of the group of letters. A, A, G, E, E, E, M

In Exercises 4144, find the number of distinguishable permutations of the group of letters. B, B, B, T, T, T, T, T

In Exercises 4144, find the number of distinguishable permutations of the group of letters. A, L, G, E, B, R, A

In Exercises 4144, find the number of distinguishable permutations of the group of letters. M, I, S, S, I, S, S, I, P, P, I

Write all permutations of the letters A, B, C, and D.

Write all permutations of the letters A, B, C, and D if the letters B and C must remain between the letters A and D.

Batting Order A baseball coach is creating a nine-player batting order by selecting from a team of 15 players. How many different batting orders are possible?

Athletics Six sprinters have qualified for the finals in the 100-meter dash at the NCAA national track meet. In how many ways can the sprinters come in first, second, and third? (Assume there are no ties.)

Jury Selection From a group of 40 people, a jury of 12 people is to be selected. In how many different ways can the jury be selected?

Committee Members As of January 2005, the U.S. Senate Committee on Indian Affairs had 14 members. Assuming party affiliation was not a factor in selection, how many different committees were possible from the 100 U.S. senators?

Write all possible selections of two letters that can be formed from the letters A, B, C, D, E, and F. (The order of the two letters is not important.)

Forming an Experimental Group In order to conduct an experiment, five students are randomly selected from a class of 20. How many different groups of five students are possible?

Lottery Choices In the Massachusetts Mass Cash game, a player chooses five distinct numbers from 1 to 35. In how many ways can a player select the five numbers?

Lottery Choices In the Louisiana Lotto game, a player chooses six distinct numbers from 1 to 40. In how many ways can a player select the six numbers?

Defective Units A shipment of 10 microwave ovens contains three defective units. In how many ways can a vending company purchase four of these units and receive (a) all good units, (b) two good units, and (c) at least two good units?

Interpersonal Relationships The complexity of interpersonal relationships increases dramatically as the size of a group increases. Determine the numbers of different two-person relationships in groups of people of sizes (a) 3, (b) 8, (c) 12, and (d) 20.

Poker Hand You are dealt five cards from an ordinary deck of 52 playing cards. In how many ways can you get (a) a full house and (b) a five-card combination containing two jacks and three aces? (A full house consists of three of one kind and two of another. For example, A-A-A-5-5 and K-K-K-10-10 are full houses.)

Job Applicants A toy manufacturer interviews eight people for four openings in the research and development department of the company. Three of the eight people are women. If all eight are qualified, in how many ways can the employer fill the four positions if (a) the selection is random and (b) exactly two selections are women?

Forming a Committee A six-member research committee at a local college is to be formed having one administrator, three faculty members, and two students. There are seven administrators, 12 faculty members, and 20 students in contention for the committee. How many six-member committees are possible?

Law Enforcement A police department uses computer imaging to create digital photographs of alleged perpetrators from eyewitness accounts. One software package contains 195 hairlines, 99 sets of eyes and eyebrows, 89 noses, 105 mouths, and 74 chins and cheek structures. (a) Find the possible number of different faces that the software could create. (b) A eyewitness can clearly recall the hairline and eyes and eyebrows of a suspect. How many different faces can be produced with this information? Geometry diagonal

In Exercises 6164, find the number of diagonals of the polygon. (A line segment connecting any two nonadjacent vertices is called a diagonal of the polygon.) Pentagon

In Exercises 6164, find the number of diagonals of the polygon. (A line segment connecting any two nonadjacent vertices is called a diagonal of the polygon.) Hexagon

In Exercises 6164, find the number of diagonals of the polygon. (A line segment connecting any two nonadjacent vertices is called a diagonal of the polygon.) Octagon

In Exercises 6164, find the number of diagonals of the polygon. (A line segment connecting any two nonadjacent vertices is called a diagonal of the polygon.) Decagon (10 sides)

Lottery Powerball is a lottery game that is operated by the Multi-State Lottery Association and is played in 27 states, Washington D.C., and the U.S. Virgin Islands. The game is played by drawing five white balls out of a drum of 53 white balls (numbered 153) and one red powerball out of a drum of 42 red balls (numbered 1 42). The jackpot is won by matching all five white balls in any order and the red powerball. (a) Find the possible number of winning Powerball numbers. (b) Find the possible number of winning Powerball numbers if the jackpot is won by matching all five white balls in order and the red power ball. (c) Compare the results of part (a) with a state lottery in which a jackpot is won by matching six balls from a drum of 53 balls.

Permutations or Combinations? Decide whether each scenario should be counted using permutations or combinations. Explain your reasoning. (a) Number of ways 10 people can line up in a row for concert tickets (b) Number of different arrangements of three types of flowers from an array of 20 types (c) Number of three-digit pin numbers for a debit card (d) Number of two-scoop ice cream cones created from 31 different flavors True or False?

True or False? In Exercises 67 and 68, determine whether the statement is true or false. Justify your answer. The number of letter pairs that can be formed in any order from any of the first 13 letters in the alphabet (AM) is an example of a permutation.

True or False? In Exercises 67 and 68, determine whether the statement is true or false. Justify your answer. The number of permutations of elements can be determined by using the Fundamental Counting Principle.

What is the relationship between and ?

Without calculating the numbers, determine which of the following is greater. Explain. (a) The number of combinations of 10 elements taken six at a time (b) The number of permutations of 10 elements taken six at a time Proof

In Exercises 7174, prove the identity.

In Exercises 7174, prove the identity.

In Exercises 7174, prove the identity.

In Exercises 7174, prove the identity.

Think About It Can your calculator evaluate If not, explain why.

Writing Explain in words the meaning of

In Exercises 7780, evaluate the function at each specified value of the independent variable and simplify. (a) (b) (c)

In Exercises 7780, evaluate the function at each specified value of the independent variable and simplify. (a) (b) (c)

In Exercises 7780, evaluate the function at each specified value of the independent variable and simplify. (a) (b) (c)

In Exercises 7780, evaluate the function at each specified value of the independent variable and simplify. (a) (b) (c)

In Exercises 8184, solve the equation. Round your answer to two decimal places, if necessary.

In Exercises 8184, solve the equation. Round your answer to two decimal places, if necessary.

In Exercises 8184, solve the equation. Round your answer to two decimal places, if necessary. log

In Exercises 8184, solve the equation. Round your answer to two decimal places, if necessary. ex_3 _ 16