Question 1

A Social Security number has nine digits. How many Social Security numbers are possible? ​

Accepted Answer

Each of the nine digits in a Social Security number can be any digit from 0 to 9, making 10 possibilities for each digit. Since there are nine digits, the total number of possible Social Security numbers is 10^9 = 1,000,000,000.

Question 2

The 2010 BMW 335i Coupe is offered with a choice of  exterior colors (  metallic and  standard),  interior colors, and  trims. How many combinations involving color and trim are available for the model? ​

Accepted Answer

The answer of The 2010 BMW 335i Coupe is offered...

Question 3

A computer dating service uses the results of its compatibility survey for arranging dates. The survey consists of 30 questions, each having five possible answers. How many different responses are possible if every question is answered? ​

Accepted Answer

The answer of A computer dating service uses the results...

Question 4

In a survey conducted by a union, members were asked to rate the importance of the following issues: (1) job security, (2) increased fringe benefits, and (3) improved working conditions. Three different responses were allowed for each issue. Among completed surveys, how many different responses to this survey were possible? ​

Accepted Answer

For each issue, there are 3 different responses possible. Since there are 3 issues, the total number of different responses possible is 3^3 = 27.

Question 5

Computers manufactured by a certain company have a serial number consisting of a letter of the alphabet followed by a five-digit number. If all the serial numbers of this type have been used, how many sets have already been manufactured? ​

Accepted Answer

The answer of Computers manufactured by a certain company have...

Question 6

A coin is tossed 4 times and the sequence of heads and tails is recorded. Use the generalized multiplication principle to determine the number of outcomes of this activity. ​

Accepted Answer

Each toss of the coin has 2 possible outcomes (heads or tails), and since there are 4 tosses, the total number of outcomes is 2^4 = 16.

Question 7

An opinion poll was conducted by the Morris Polling Group. Respondents were classified according to their sex (M or F), political affiliation (D, I, R), and the region of the country in which they reside (NW, W, C, S). ​
Use the generalized multiplication principle to determine the number of possible classifications.
​

Accepted Answer

There are 2 options for sex (M or F), 3 options for political affiliation (D, I, R), and 4 options for the region (NW, W, C, S). Using the generalized multiplication principle, the total number of possible classifications is 2 * 3 * 4 = 24.

Question 8

An exam consists of nine true-or-false questions. Assuming that every question is answered, in how many different ways can a student complete the exam? In how many ways may the exam be completed if a penalty is imposed for each incorrect answer, so that a student may leave some questions unanswered? ​

Accepted Answer

For each question, there are 2 choices (true or false), so there are $2^9 = 512$ ways to answer all the questions.
If there is a penalty for incorrect answers, there are 3 choices for each question (true, false, or leave unanswered), so there are $3^9 = 19,683$ ways to answer the exam.

Question 9

Four soups, five entrees, and four desserts are listed on the ''Special'' menu at the Neptune Restaurant. How many different selections consisting of one soup, one entree, and one dessert can a customer choose from this menu? ​

Accepted Answer

The number of different selections is found by multiplying the number of options for each course together: 4 soups * 5 entrees * 4 desserts = 80 different selections.

Question 10

Four commuter trains and two express buses depart from city A to city B in the morning, and three commuter trains and three express buses operate on the return trip in the evening. In how many ways can a commuter from city A to city B complete a daily round trip via bus and/or train? ​

Accepted Answer

The answer of Four commuter trains and two express buses...

Question 11

A new state employee is offered a choice of ten basic health plans, three dental plans, and three vision care plans. How many different health-care plans are there to choose from if one plan is selected from each category? ​

Accepted Answer

The answer of A new state employee is offered a...

Question 12

To gain access to his account, a customer using an automatic machine (ATM) must enter a five-digit code. If repetition of same five digit is not allowed (for example, 55555), how many possible combinations are there?

Accepted Answer

The total number of five-digit codes is 10^5 = 100,000. Subtract the 10 codes where all digits are the same (00000, 11111, ..., 99999), resulting in 100,000 - 10 = 99,990 possible combinations.

Question 13

The number of three-digit numbers that can be formed from the digits 1, 2, 3, and 4 if the number must be even is 19.

Accepted Answer

If a number is even, its last digit must be 2 or 4. 
- If the last digit is 2, there are 3 choices for the first digit (1, 3, or 4) and 2 choices for the second digit (the remaining digits), giving a total of 3 x 2 = 6 numbers.
- If the last digit is 4, there are again 3 choices for the first digit and 2 choices for the second digit, giving another 6 numbers.
Therefore, there are only 12 numbers that can be formed, not 19.

Question 14

An automobile manufacturer has four different subcompact cars in the line. Customers selecting one of these cars have a choice of two engine sizes, five body styles, and two color schemes. How many different selections can a customer make? ​

Accepted Answer

The total number of different selections a customer can make is calculated by multiplying the number of options in each category. Thus, it's 4 (cars) * 2 (engine sizes) * 5 (body styles) * 2 (color schemes) = 80 different selections.

Question 15

A female executive selecting her wardrobe purchased three blazers, five blouses, and five skirts in coordinating colors. How many ensembles consisting of a blazer, a blouse, and a skirt can she create from this collection? ​

Accepted Answer

The number of ensembles can be calculated by multiplying the number of options for each clothing item: 3 blazers * 5 blouses * 5 skirts = 75 possible ensembles.

Question 16

How many four-letter code words can be constructed from the first ten letters of the Greek alphabet if no repetitions allowed? ​

Accepted Answer

The first ten letters of the Greek alphabet can be used to form 4-letter code words without repetition. This is a permutation problem, calculated as 10P4 = 10! / (10-4)! = 10 * 9 * 8 * 7 = 5,040 code words.

Question 17

A warranty identification number for a certain product consists of a letter of the alphabet followed by a four-digit number. How many possible identification numbers are there if the first digit of the four-digit number must be nonzero? ​

Accepted Answer

The first part of the identification number is a letter of the alphabet, so there are 26 possible choices for this part. The second part is a four-digit number where the first digit must be nonzero. Since there are 9 choices for the first digit (1-9), 10 choices for each of the next three digits (0-9), the total number of possibilities for the four-digit number is 9 * 10 * 10 * 10 = 9,000. Therefore, the total number of possible identification numbers is 26 * 9,000 = 234,000.

Question 18

If there are four toppings available, then the number of different pizzas that can be made is  , or 128, different pizzas. ​

Accepted Answer

The answer of If there are four toppings available, then...

Question 19

In recent years, the state of California issued license plates using a combination of two letters of the alphabet followed by two digits, followed by another three letters of the alphabet. How many different license plates can be issued using this configuration?

Accepted Answer

There are 26 choices for each of the first two letters, and 10 choices for each of the two digits, and 26 choices for each of the last three letters. So, the total number of different license plates is $26\cdot 26\cdot 10\cdot 10\cdot 26\cdot 26=1,\!188,\!137,\!600$. Therefore, the best choice is (A).

Question 20

In a state lottery there are 12 finalists eligible for the Big Money Draw. In how many ways can the first, second, and third prizes be awarded if no ticket holder may win more than one prize? ​

Accepted Answer

This is a permutation problem because the order in which the prizes are awarded matters and no ticket holder can win more than one prize. The formula for permutations is nPr = n! / (n-r)!, where n is the total number of items and r is the number of items to choose. Here, n=12 (finalists) and r=3 (prizes), so the calculation is 12P3 = 12! / (12-3)! = 12! / 9! = 12 x 11 x 10 = 1,320 different selections.

Quiz 6: Sets and Counting

A Social Security number has nine digits. How many Social Security numbers are possible?

The 2010 BMW 335i Coupe is offered with a choice of exterior colors ( metallic and standard) , interior colors, and trims. How many combinations involving color and trim are available for the model?

A computer dating service uses the results of its compatibility survey for arranging dates. The survey consists of 30 questions, each having five possible answers. How many different responses are possible if every question is answered?

Computers manufactured by a certain company have a serial number consisting of a letter of the alphabet followed by a five-digit number. If all the serial numbers of this type have been used, how many sets have already been manufactured?

A coin is tossed 4 times and the sequence of heads and tails is recorded. Use the generalized multiplication principle to determine the number of outcomes of this activity.

An exam consists of nine true-or-false questions. Assuming that every question is answered, in how many different ways can a student complete the exam? In how many ways may the exam be completed if a penalty is imposed for each incorrect answer, so that a student may leave some questions unanswered?

Four soups, five entrees, and four desserts are listed on the ''Special'' menu at the Neptune Restaurant. How many different selections consisting of one soup, one entree, and one dessert can a customer choose from this menu?

Four commuter trains and two express buses depart from city A to city B in the morning, and three commuter trains and three express buses operate on the return trip in the evening. In how many ways can a commuter from city A to city B complete a daily round trip via bus and/or train?

A new state employee is offered a choice of ten basic health plans, three dental plans, and three vision care plans. How many different health-care plans are there to choose from if one plan is selected from each category?

To gain access to his account, a customer using an automatic machine (ATM) must enter a five-digit code. If repetition of same five digit is not allowed (for example, 55555) , how many possible combinations are there?

The number of three-digit numbers that can be formed from the digits 1, 2, 3, and 4 if the number must be even is 19.

An automobile manufacturer has four different subcompact cars in the line. Customers selecting one of these cars have a choice of two engine sizes, five body styles, and two color schemes. How many different selections can a customer make?

A female executive selecting her wardrobe purchased three blazers, five blouses, and five skirts in coordinating colors. How many ensembles consisting of a blazer, a blouse, and a skirt can she create from this collection?

How many four-letter code words can be constructed from the first ten letters of the Greek alphabet if no repetitions allowed?

A warranty identification number for a certain product consists of a letter of the alphabet followed by a four-digit number. How many possible identification numbers are there if the first digit of the four-digit number must be nonzero?

If there are four toppings available, then the number of different pizzas that can be made is , or 128, different pizzas.

In recent years, the state of California issued license plates using a combination of two letters of the alphabet followed by two digits, followed by another three letters of the alphabet. How many different license plates can be issued using this configuration?

In a state lottery there are 12 finalists eligible for the Big Money Draw. In how many ways can the first, second, and third prizes be awarded if no ticket holder may win more than one prize?

Quiz 6: Sets and Counting

A Social Security number has nine digits. How many Social Security numbers are possible? ​

The 2010 BMW 335i Coupe is offered with a choice of exterior colors ( metallic and standard) , interior colors, and trims. How many combinations involving color and trim are available for the model? ​

A computer dating service uses the results of its compatibility survey for arranging dates. The survey consists of 30 questions, each having five possible answers. How many different responses are possible if every question is answered? ​

Computers manufactured by a certain company have a serial number consisting of a letter of the alphabet followed by a five-digit number. If all the serial numbers of this type have been used, how many sets have already been manufactured? ​

A coin is tossed 4 times and the sequence of heads and tails is recorded. Use the generalized multiplication principle to determine the number of outcomes of this activity. ​

Four soups, five entrees, and four desserts are listed on the ''Special'' menu at the Neptune Restaurant. How many different selections consisting of one soup, one entree, and one dessert can a customer choose from this menu? ​

Four commuter trains and two express buses depart from city A to city B in the morning, and three commuter trains and three express buses operate on the return trip in the evening. In how many ways can a commuter from city A to city B complete a daily round trip via bus and/or train? ​

A new state employee is offered a choice of ten basic health plans, three dental plans, and three vision care plans. How many different health-care plans are there to choose from if one plan is selected from each category? ​

To gain access to his account, a customer using an automatic machine (ATM) must enter a five-digit code. If repetition of same five digit is not allowed (for example, 55555) , how many possible combinations are there?

The number of three-digit numbers that can be formed from the digits 1, 2, 3, and 4 if the number must be even is 19.

An automobile manufacturer has four different subcompact cars in the line. Customers selecting one of these cars have a choice of two engine sizes, five body styles, and two color schemes. How many different selections can a customer make? ​

A female executive selecting her wardrobe purchased three blazers, five blouses, and five skirts in coordinating colors. How many ensembles consisting of a blazer, a blouse, and a skirt can she create from this collection? ​

How many four-letter code words can be constructed from the first ten letters of the Greek alphabet if no repetitions allowed? ​

A warranty identification number for a certain product consists of a letter of the alphabet followed by a four-digit number. How many possible identification numbers are there if the first digit of the four-digit number must be nonzero? ​

If there are four toppings available, then the number of different pizzas that can be made is , or 128, different pizzas. ​

In recent years, the state of California issued license plates using a combination of two letters of the alphabet followed by two digits, followed by another three letters of the alphabet. How many different license plates can be issued using this configuration?

In a state lottery there are 12 finalists eligible for the Big Money Draw. In how many ways can the first, second, and third prizes be awarded if no ticket holder may win more than one prize? ​

A Social Security number has nine digits. How many Social Security numbers are possible?

The 2010 BMW 335i Coupe is offered with a choice of exterior colors ( metallic and standard) , interior colors, and trims. How many combinations involving color and trim are available for the model?

A computer dating service uses the results of its compatibility survey for arranging dates. The survey consists of 30 questions, each having five possible answers. How many different responses are possible if every question is answered?

Computers manufactured by a certain company have a serial number consisting of a letter of the alphabet followed by a five-digit number. If all the serial numbers of this type have been used, how many sets have already been manufactured?

A coin is tossed 4 times and the sequence of heads and tails is recorded. Use the generalized multiplication principle to determine the number of outcomes of this activity.

Four soups, five entrees, and four desserts are listed on the ''Special'' menu at the Neptune Restaurant. How many different selections consisting of one soup, one entree, and one dessert can a customer choose from this menu?

Four commuter trains and two express buses depart from city A to city B in the morning, and three commuter trains and three express buses operate on the return trip in the evening. In how many ways can a commuter from city A to city B complete a daily round trip via bus and/or train?

A new state employee is offered a choice of ten basic health plans, three dental plans, and three vision care plans. How many different health-care plans are there to choose from if one plan is selected from each category?

An automobile manufacturer has four different subcompact cars in the line. Customers selecting one of these cars have a choice of two engine sizes, five body styles, and two color schemes. How many different selections can a customer make?

A female executive selecting her wardrobe purchased three blazers, five blouses, and five skirts in coordinating colors. How many ensembles consisting of a blazer, a blouse, and a skirt can she create from this collection?

How many four-letter code words can be constructed from the first ten letters of the Greek alphabet if no repetitions allowed?

A warranty identification number for a certain product consists of a letter of the alphabet followed by a four-digit number. How many possible identification numbers are there if the first digit of the four-digit number must be nonzero?

If there are four toppings available, then the number of different pizzas that can be made is , or 128, different pizzas.

In a state lottery there are 12 finalists eligible for the Big Money Draw. In how many ways can the first, second, and third prizes be awarded if no ticket holder may win more than one prize?