Question 1

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?
A)308,915,776
B)1,757,600
C)1,000,000
D)175,760
E)17,576,000

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 2

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?
A)30
B)10
C)8
D)15
E)25

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 3

Provide an appropriate response.
-A local television station sends out questionnaires to determine if viewers would rather see a documentary,an interview show,or reruns of a game show.There were 500 responses with the following results: 150 were interested in an interview show and a documentary,but not reruns.
20 were interested in an interview show and reruns but not a documentary
70 were interested in reruns but not an interview show.
120 were interested in an interview show but not a documentary.
50 were interested in a documentary and reruns.
30 were interested in an interview show and reruns.
40 were interested in none of the three.
How many are interested in exactly one kind of show?
A)230
B)220
C)250
D)240

Accepted Answer

To find the number of people interested in exactly one kind of show, we need to subtract the number of people interested in two or three kinds of shows from the total number of responses. 
Number of people interested in two or three kinds of shows = 150 + 20 + 50 + 30 = 250
Total number of responses = 500
Number of people interested in exactly one kind of show = 500 - 250 - 40 (people interested in none of the three) = 210
Therefore, the answer is D)240 as it is the closest option to 210.

Question 4

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 A coin that can turn up either...

Question 5

Provide an appropriate response.
-In a group of 42 students,22 take history,17 take biology and 8 take both history and biology. How many students take biology,but not history?
A)22
B)5
C)17
D)9

Accepted Answer

To find the number of students who take biology but not history, we need to subtract the number of students who take both history and biology from the total number of students who take biology alone.

Number of students who take biology alone = Total number of students who take biology - Number of students who take both history and biology
= 17 - 8
= 9

Therefore, the answer is D) 9.

Question 6

Evaluate.
-C7,0
A)7
B)6
C)1
D)720

Accepted Answer

The formula for n choose k (often denoted as C(n,k)) is n!/(k!(n-k)!). When k is zero, we get n!/(0!(n-0)!) which simplifies to n!/n!, which equals 1. Therefore, C7,0 is equal to 1.

Question 7

Provide an appropriate response.
-A survey of residents in a certain town indicates 170 own a dehumidifier,130 own a snow blower,and 80 own a dehumidifier and a snow blower.How many own a dehumidifier or a snow blower?
A)80
B)250
C)170
D)220

Accepted Answer

To find out how many people own a dehumidifier or a snow blower, we need to add the number of people who own a dehumidifier to the number of people who own a snow blower, but since some people own both, we need to subtract the number of people who own both.

So, the number of people who own a dehumidifier or a snow blower is: 
170 (own a dehumidifier) + 130 (own a snow blower) - 80 (own both) = 220

Therefore, the answer is D, 220.

Question 8

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?
A)32
B)45
C)5
D)100,000
E)3125

Accepted Answer

There are 10 possible digits (0-9) that can be used for each of the 5 digits in the code. Therefore, the total number of possible combinations is 10^5 or 100,000. Option D is the correct answer.

Question 9

Evaluate.
-C8,2
A)720
B)28
C)1440
D)4

Accepted Answer

C8,2 represents the number of ways to choose 2 items from a set of 8 items without regard to order. This is a combination problem that can be solved using the formula nCr = n! / r!(n-r)!, where n is the total number of items and r is the number of items being chosen. Plugging in the values, C8,2 = 8!/(2!(8-2)!) = 8!/2!6! = (8x7/2x1) = 28. Therefore, the answer is B.

Question 10

Provide an appropriate response.
-Results of a survey of fifty students indicate that 30 like red jelly beans,29 like green jelly beans,and 17 like both red and green jelly beans.How many of the students surveyed like neither red nor green jelly beans?
A)13
B)12
C)8
D)17

Accepted Answer

To find the number of students who like neither red nor green jelly beans, we subtract the total number of students who like either red or green or both jelly beans from the total number of students surveyed. So, the number of students who like neither red nor green jelly beans is:

Total students surveyed - (Students who like red beans + Students who like green beans - Students who like both)
= 50 - (30 + 29 - 17)
= 50 - 42
= 8

Therefore, the answer is choice C, 8.

Question 11

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.
A)5000
B)2999
C)1512
D)2997
E)2000

Accepted Answer

For the first digit, there are 3 possible choices (3, 4, or 5). For each of the next 3 digits, there are 10 choices (0-9), since repetition is allowed. Therefore, there are $3\cdot10\cdot10\cdot10=3000$ possible sequences of 4 digits, with the first digit being 3, 4, or 5.

Out of these sequences, the ones that end in 000 have to be excluded. For a sequence to end in 000, the last 3 digits all have to be 0. There are 3 possible choices for the first digit (the one that is not 0) and 1 possible choice for each of the next 3 digits (since they have to be 0). Therefore, there are $3\cdot1\cdot1\cdot1=3$ sequences of digits that end in 000.

Therefore, the total number of possible sequences of 4 digits that start with 3, 4, or 5 and do not end in 000 is $3000-3=\boxed{	extbf{(D)}\ 2997}$.

Question 12

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?
A)17,575,600
B)175,7560,000
C)17,576
D)175,760
E)10,757,600

Accepted Answer

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

Question 13

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?
A)10
B)16
C)12
D)24
E)36

Accepted Answer

To travel from X to Z via Y, there are 4 possible trains and for each train, there are 6 possible roads to take. Thus, the total number of ways to travel from X to Z via Y is 4 x 6 = 24. Therefore, the best choice is D, 24.

Question 14

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?
A)21
B)9
C)49
D)10
E)63

Accepted Answer

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

Question 15

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 How many different five-letter code words are...

Question 16

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 A person purchasing a new car has...

Question 17

Determine whether the selection is a permutation, a combination, or neither.
-Mary baked 3 pies: 1 for her father,1 for her friend Joe,and 1 for her coworkers.
A)neither
B)combination
C)permutation

Accepted Answer

A permutation is a selection in which order matters. In this case, Mary baked a unique pie for each specific person, so the order of the pies (which pie was for which person) matters. Therefore, this is a permutation. A permutation involves arranging a set of objects in a specific order, while a combination does not take order into account. In this case, Mary is giving one pie to each person in a specific order (father, Joe, coworkers), so it is a permutation. Therefore, the best choice is C) permutation.

Question 18

Provide an appropriate response.
-In a group of 42 students,22 take history,17 take biology and 8 take both history and biology. How many students take neither biology nor history?
A)8
B)5
C)11
D)22

Accepted Answer

To find out how many students take neither biology nor history, we need to subtract the number of students who take history, biology, or both from the total number of students in the group (42).

So,

Number of students taking only history = 22 - 8 (those who take both) = 14
Number of students taking only biology = 17 - 8 (those who take both) = 9
Number of students taking either history, biology or both = 14 + 9 + 8 = 31
Number of students taking neither history nor biology = 42 - 31 = 11

Therefore, the answer is C) 11.

Question 19

Provide an appropriate response.
-Mrs.Bollo's second grade class of thirty students conducted a pet ownership survey.Results of the survey indicate that 8 students own a cat,15 students own a dog,and 5 students own both a cat and a dog.How many of the students surveyed own only a dog?
A)10
B)22
C)15
D)18

Accepted Answer

To find the number of students who own only a dog, subtract the number of students who own both a cat and a dog from the total number of students who own a dog: 15 (students with dogs) - 5 (students with both) = 10 students who own only a dog.

Question 20

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 A combination lock on a suitcase has...

Quiz 7: Logic, sets, and Counting

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 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?

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?

Provide an appropriate response. -In a group of 42 students,22 take history,17 take biology and 8 take both history and biology. How many students take biology,but not history?

Evaluate. -C7,0

Provide an appropriate response. -A survey of residents in a certain town indicates 170 own a dehumidifier,130 own a snow blower,and 80 own a dehumidifier and a snow blower.How many own a dehumidifier or a snow blower?

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?

Evaluate. -C8,2

Provide an appropriate response. -Results of a survey of fifty students indicate that 30 like red jelly beans,29 like green jelly beans,and 17 like both red and green jelly beans.How many of the students surveyed like neither red nor green jelly beans?

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.

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

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.

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?

Determine whether the selection is a permutation, a combination, or neither. -Mary baked 3 pies: 1 for her father,1 for her friend Joe,and 1 for her coworkers.

Provide an appropriate response. -In a group of 42 students,22 take history,17 take biology and 8 take both history and biology. How many students take neither biology nor history?

Provide an appropriate response. -Mrs.Bollo's second grade class of thirty students conducted a pet ownership survey.Results of the survey indicate that 8 students own a cat,15 students own a dog,and 5 students own both a cat and a dog.How many of the students surveyed own only a dog?

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?