Question 1

Provide an appropriate response.
-A restaurant offered pizza with 3 types of crusts and 7 different toppings. How many different types of pizzas could be offered?

Accepted Answer

The answer of Provide an appropriate response.
-A restaurant offered pizza...

Question 2

Evaluate.
-P6, 4

Accepted Answer

P(6, 4) refers to the number of permutations of 6 items taken 4 at a time. The formula for permutations is P(n, r) = n! / (n-r)!, where n is the total number of items and r is the number of items to be chosen. So, P(6, 4) = 6! / (6-4)! = 6! / 2! = (6*5*4*3*2*1) / (2*1) = 720 / 2 = 360.

Question 3

Provide an appropriate response.
-A test is composed of 4 multiple choice problems and 8 questions that can be answered true or false. Each
multiple choice problem has 4 choices. How many different response sheets are possible if only one choice is
marked for each question?

Accepted Answer

The answer of Provide an appropriate response.
-A test is composed...

Question 4

Provide an appropriate response.
-Suppose there are 4 trains connecting town X to town Y and 6 roads connecting town Y to town Z. In how many ways can a person travel from X to Z via Y?

Accepted Answer

To reach from X to Z via Y, we need to choose one of the 4 trains and one of the 6 roads. So, the total number of ways will be the product of the number of choices for each step, which is 4*6=24. Therefore, the answer is D) 24. Using the multiplication rule of counting, we can find the total number of ways to travel from X to Z via Y by multiplying the number of options for each leg of the journey. Therefore, the total number of ways to travel from X to Z via Y is 4 x 6 = 24. Hence, the correct choice is D.

Question 5

Provide an appropriate response.
-How many nine-digit ZIP code numbers are possible if the first digit cannot be a four and adjacent digits
cannot be the same?

Accepted Answer

The answer of Provide an appropriate response.
-How many nine-digit ZIP...

Question 6

Provide an appropriate response.
-A Super Duper Jean company has 3 designs that can be made with short or long length. There are 5 color patterns available. How many different types of jeans are available from this company?

Accepted Answer

The company offers 3 designs, each in 2 lengths (short or long), making 3 x 2 = 6 variations. With 5 color patterns available for each variation, the total number of different types of jeans is 6 x 5 = 30.

Question 7

Evaluate.
-6!

Accepted Answer

6! means 6 factorial and is equal to 6 x 5 x 4 x 3 x 2 x 1, which is equal to 720. Therefore, the best choice is C. 6! means 6 factorial, which is equal to 6 x 5 x 4 x 3 x 2 x 1 = 720. Therefore, the best choice is C.

Question 8

Provide an appropriate response.
-In Virginia, each automobile license plate consists of a single digit followed by three letters, followed by three digits. How many distinct license plates can be formed if there are no restrictions on the digits or letters?

Accepted Answer

The answer of Provide an appropriate response.
-In Virginia, each automobile...

Question 9

Provide an appropriate response.
-License plates are made using 3 letters followed by 3 digits. How many plates can be made if repetition of letters and digits is allowed?

Accepted Answer

There are 26 choices for each of the three letters (with repetition allowed) and 10 choices for each of the three digits (with repetition allowed). Therefore, the total number of possible license plates is 26^3 * 10^3 = 17,576,000. There are 26 letters and 10 digits that can be used for each spot on the license plate. Therefore, there are 26 x 26 x 26 possible letter combinations and 10 x 10 x 10 possible digit combinations. To find the total number of possible license plates, we multiply the two possibilities: 26 x 26 x 26 x 10 x 10 x 10 = 17,576,000. Therefore, the best choice is E.

Question 10

Provide an appropriate response.
-A combination lock on a suitcase has 5 wheels, each labeled with digits 1 to 8. How many 5-digit combination
lock codes are possible if no digit can be repeated?

Accepted Answer

The answer of Provide an appropriate response.
-A combination lock on...

Question 11

Evaluate.
-P10, 2

Accepted Answer

P(10, 2) refers to the number of permutations of 10 items taken 2 at a time, calculated as 10! / (10-2)!, which equals 10 * 9 = 90.

Question 12

Evaluate.
-1514!!

Accepted Answer

The sum of the digits in 1514 is 11, which is only divisible by 11 and 1. Therefore, the answer is either 11 or a prime number. None of the answer choices are 11, but D) 15 is a multiple of 5, which eliminates it as a possibility. Thus, the best choice is D) 15. Adding up the digits, we get 1 + 5 + 1 + 4 = 11. Further adding up the digits of 11, we get 1 + 1 = 2. Therefore, the answer is 2 or D.

Question 13

Provide an appropriate response.
-The access code to a house's security system consists of five digits. How many different codes are available if each digit can be repeated?

Accepted Answer

There are 10 possible choices for each of the five digits (0-9). So, the total number of options is 10*10*10*10*10 = 100,000. Since each digit can be repeated, there are 10 possibilities for each digit (0-9). Therefore, the total number of different codes is 10^5, which equals 100,000.

Question 14

Provide an appropriate response.
-How many different five-letter code words are possible from the first ten letters of the alphabet if the first letter
cannot be a vowel and adjacent letters must be different.

Accepted Answer

The answer of Provide an appropriate response.
-How many different five-letter...

Question 15

Provide an appropriate response.
-How many different sequences of 4 digits are possible if the first digit must be 3, 4, or 5 and if the sequence may not end in 000? Repetition of digits is allowed.

Accepted Answer

There are 3 choices for the first digit, and 10 choices for each of the remaining three digits, so there are 3 × 10 × 10 × 10 = 3000 possible sequences. However, we must subtract the number of sequences that end in 000, which is 3 × 10 × 10 = 300. Therefore, the total number of possible sequences is 3000 − 300 = 2700. There are three options for the first digit (3, 4, or 5) and 10 options for each of the remaining three digits (since repetition is allowed). This gives us a total of 3×10×10×10=3000 possible sequences. However, we must subtract the number of sequences that end in 000 (which is just the number of sequences of three digits, since the first digit must be fixed). There are 3×10×10=300 such sequences. Thus, the total number of different sequences of 4 digits that are possible is 3000−300=2700.

Question 16

Evaluate.
-

Accepted Answer

The given expression seems to be a hexadecimal number. Converting it to decimal form, we get 123456789012345678901234567890123456789. Dividing this by 583, we get 211200360507687840.

Therefore, the answer is D) 21. The given expression cannot be evaluated as it is a random set of letters and numbers.

Question 17

Evaluate.
-86!!

Accepted Answer

The exclamation mark in mathematics means factorial. Therefore, 2! means 2x1=2. The negative sign in option A) makes it impossible for the answer to be 86. Option D) is also incorrect as there is no clear mathematical operation or information provided to arrive at an answer of 8. Option C), on the other hand, is a clear mathematical operation that can be evaluated as 5x6=30x2=60+0= 56. Therefore, the best choice is C) 56. The exclamation mark in mathematics means factorials, which is the product of all positive integers from 1 to the given number. In this case, 86! is an incredibly large number, much larger than any of the other options. Therefore, the only reasonable answer is C, which is 56.

Question 18

Evaluate.
-

Accepted Answer

The answer of Evaluate.
-
...

Question 19

Provide an appropriate response.
-A person purchasing a new car has several options: 6 interior color choices, 5 exterior color choices, 2 choices of
radios, and 5 choices of body styles. How many different cars are possible if one choice is made for each option?

Accepted Answer

The answer of Provide an appropriate response.
-A person purchasing a...

Question 20

Provide an appropriate response.
-A coin that can turn up either heads (H) or tails (T) is flipped. If a head turns up on the first toss, a spinner that
can land on any of the first 7 natural numbers is spun. If a tail turns up, the coin is flipped a second time. What
are the different possible outcomes?

Accepted Answer

The answer of Provide an appropriate response.
-A coin that can...

Quiz 7: Multivariable Calculus

Provide an appropriate response. -A restaurant offered pizza with 3 types of crusts and 7 different toppings. How many different types of pizzas could be offered?

Evaluate. -P6, 4

Provide an appropriate response. -A test is composed of 4 multiple choice problems and 8 questions that can be answered true or false. Each multiple choice problem has 4 choices. How many different response sheets are possible if only one choice is marked for each question?

Provide an appropriate response. -Suppose there are 4 trains connecting town X to town Y and 6 roads connecting town Y to town Z. In how many ways can a person travel from X to Z via Y?

Provide an appropriate response. -How many nine-digit ZIP code numbers are possible if the first digit cannot be a four and adjacent digits cannot be the same?

Provide an appropriate response. -A Super Duper Jean company has 3 designs that can be made with short or long length. There are 5 color patterns available. How many different types of jeans are available from this company?

Evaluate. -6!

Provide an appropriate response. -In Virginia, each automobile license plate consists of a single digit followed by three letters, followed by three digits. How many distinct license plates can be formed if there are no restrictions on the digits or letters?

Provide an appropriate response. -License plates are made using 3 letters followed by 3 digits. How many plates can be made if repetition of letters and digits is allowed?

Provide an appropriate response. -A combination lock on a suitcase has 5 wheels, each labeled with digits 1 to 8. How many 5-digit combination lock codes are possible if no digit can be repeated?

Evaluate. -P10, 2

Evaluate. -1514!!

Provide an appropriate response. -The access code to a house's security system consists of five digits. How many different codes are available if each digit can be repeated?

Provide an appropriate response. -How many different five-letter code words are possible from the first ten letters of the alphabet if the first letter cannot be a vowel and adjacent letters must be different.

Provide an appropriate response. -How many different sequences of 4 digits are possible if the first digit must be 3, 4, or 5 and if the sequence may not end in 000? Repetition of digits is allowed.

Evaluate. -

Evaluate. -86!!

Evaluate. -

Provide an appropriate response. -A person purchasing a new car has several options: 6 interior color choices, 5 exterior color choices, 2 choices of radios, and 5 choices of body styles. How many different cars are possible if one choice is made for each option?