Question 1

Solve the problem.
-In how many ways can 8 people be lined up?

Accepted Answer

To line up 8 people, you have 8 choices for the first person, 7 for the second, 6 for the third, and so on. Therefore, the number of arrangements is 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40,320. The number of permutations of n objects is given by n!, where ! denotes factorial. Therefore, the number of ways to line up 8 people is 8! = 40,320.

Question 2

Solve the problem.
-A man has 7 shirts and 11 ties. How many different shirt and tie arrangements can he wear?

Accepted Answer

The number of different shirt and tie arrangements can be found by multiplying the number of shirts by the number of ties, which is 7 shirts * 11 ties = 77 arrangements.

Question 3

Find the value of the permutation.
-P(5, 1)

Accepted Answer

The value of a permutation with n objects taken r at a time, denoted by P(n, r), is given by:
P(n,r) = n! / (n-r)!
So, P(5,1) = 5! / (5-1)! = 5 * 4 * 3 * 2 * 1 / 4 * 3 * 2 * 1 = 5
Therefore, the correct answer is (A) 5.

Question 4

Solve the problem.
-How many different 5-letter codes are there if only the letters A, B, C, D, E, F, G, H, and I can be used andno letter can be used more than once?

Accepted Answer

We need to find the number of permutations of 9 different items taken 5 at a time. This can be calculated as 9P5 which is equal to 9!/4! = 15,120. Therefore, there are 15,120 different 5-letter codes that can be formed.

Question 5

Find the value of the permutation.
-P(5, 1)

Accepted Answer

The value of the permutation P(5, 1) is calculated as 5! / (5-1)! = 5! / 4! = 5.

Question 6

Solve the problem.
-How many 3-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 if the first digitcannot be 0? Repeated digits are allowed.

Accepted Answer

For a 3-digit number, the first digit can be any of the 9 options (1-9), the second digit can be any of the 10 options (0-9), and the third digit can also be any of the 10 options (0-9). Therefore, the total number of possible 3-digit numbers is 9 * 10 * 10 = 900.

Question 7

Find the value of the permutation.
-P(4, 0)

Accepted Answer

The formula for permutation is:
P(n,r) = n! / (n-r)!
When r=0, the formula becomes:
P(n,0) = n! / n!
Since any number divided by itself is 1, we get:
P(n,0) = 1
Therefore, the value of P(4,0) is 1. The answer is choice A. The value of a permutation, P(n,r), represents the number of ways to choose r items from a set of n items, without replacement and where order matters. 
In this case, P(4,0) means choosing 0 items from a set of 4 items with order mattering. This can only be done in one way, which is choosing nothing at all. Therefore, P(4,0) = 1. Option A is the best choice.

Question 8

Solve the problem.
-9 different books are to be arranged on a shelf. How many different arrangements are possible?

Accepted Answer

The number of ways to arrange 9 different books on a shelf is given by 9 factorial (9!), which is 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 362,880.

Question 9

Find the value of the permutation.
-P(9, 9)

Accepted Answer

The value of the permutation P(9, 9) is calculated as 9! (9 factorial), which equals 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 362,880.

Question 10

Solve the problem.
-How many different license plates can be made using 3 letters followed by 2 digits selected from the digits0 through 9, if letters and digits may be repeated?

Accepted Answer

There are 26 choices for each of the 3 letters, and 10 choices for each of the 2 digits. Therefore, the total number of possible license plates is:
26 × 26 × 26 × 10 × 10 = 1,757,600. There are 26 choices for each of the three letters and 10 choices for each of the two digits. Therefore, the total number of possible license plates is:
26 × 26 × 26 × 10 × 10 = 1,757,600

Question 11

Solve the problem.
-How many 7-symbol codes can be formed using 4 different symbols? Repeated symbols are allowed.

Accepted Answer

Each symbol can be chosen in 4 ways. As repetitions are allowed, each of the 7 symbols can be chosen in 4 ways. Thus, the total number of 7-symbol codes that can be formed using 4 different symbols is:
4 x 4 x 4 x 4 x 4 x 4 x 4 = 16,384. Therefore, the best choice is A). There are 4 different symbols and each symbol can be used 7 times, with repetition being allowed. Therefore, the total number of codes that can be formed is:

4 x 4 x 4 x 4 x 4 x 4 x 4 = 16,384

Therefore, the answer is (A) 16,384.

Question 12

Solve the problem.
-The following data represent the marital status of females 18 years and older in a certain U.S. city.  Determine the number of females 18 years old and older who are married or widowed.

Accepted Answer

The answer of Solve the problem.
-The following data represent the...

Question 13

Solve the problem.
-A restaurant offers a choice of 4 salads, 7 main courses, and 2 desserts. How many possible 3-coursemeals are there?

Accepted Answer

The number of possible 3-course meals is 4 * 7 * 2 = 56.

Question 14

Solve the problem.
-How many arrangements of answers are possible in a multiple-choice test with 9 questions, each of whichhas 3 possible answers?

Accepted Answer

For each question, there are 3 possible answers. Therefore, there are
3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 = 3^9 = 19,683
possible arrangements of answers. For each of the 9 questions, there are 3 possible answers. Therefore, the total number of arrangements of answers is 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 = 19,683. Therefore, the correct answer is (A) 19,683.

Question 15

Solve the problem.
-List all the ordered arrangements of 4 objects 1, 2, 3, and 4 choosing 3 at a time without repetition. What isP(4, 3)?

Accepted Answer

The permutations of 4 objects taken 3 at a time is calculated as P(4, 3) = 4! / (4-3)! = 24, matching the list provided in option A.

Question 16

Solve the problem.
-A certain mathematics test consists of 20 questions. Goldie decides to answer the questions withoutreading them. In how many ways can Goldie fill in the answer sheet if the possible answers are true andfalse?

Accepted Answer

For each question, Goldie has 2 choices (true or false), so the total number of ways she can answer all 20 questions is 2^20, which is 1,048,576. Therefore, the answer is A. For each question, Goldie has two choices (true or false). Since there are 20 questions, the total number of ways Goldie can fill in the answer sheet is 2 x 2 x 2 ... (20 times) = 2^20 = 1,048,576. Therefore, the best choice is A.

Question 17

Solve the problem.
-How many 2-letter codes can be formed using the letters A, B, C, D, E, F, G, H, and I. Repeated letters areallowed.

Accepted Answer

Since repeated letters are allowed, for each of the 2 positions in the code, any of the 9 letters can be chosen, resulting in 9 * 9 = 81 possible codes.

Question 18

Solve the problem.
-Lisa has 4 skirts, 8 blouses, and 2 jackets. How many 3-piece outfits can she put together assuming anypiece goes with any other?

Accepted Answer

Lisa can create 64 possible outfits by multiplying the number of choices for each piece of clothing: 4 skirts * 8 blouses * 2 jackets = 64.

Question 19

Solve the problem.
-List all the ordered arrangements of 6 objects a, b, c, d, e, and f choosing 2 at a time without repetition.What is P(6, 2)?

Accepted Answer

P(6, 2) calculates the number of permutations of 6 objects taken 2 at a time, which equals 6! / (6-2)! = 6*5 = 30, matching the arrangements listed in option A.

Question 20

Solve the problem.
-A student must choose 1 of 4 mathematics electives, 1 of 6 science electives, and 1 of 9 programmingelectives. How many possible course selections are there?

Accepted Answer

The total number of possible course selections is found by multiplying the number of options for each elective: 4 (math) * 6 (science) * 9 (programming) = 216.

Quiz 10: Counting and Probability

Solve the problem. -In how many ways can 8 people be lined up?

Solve the problem. -A man has 7 shirts and 11 ties. How many different shirt and tie arrangements can he wear?

Find the value of the permutation. -P(5, 1)

Solve the problem. -How many different 5-letter codes are there if only the letters A, B, C, D, E, F, G, H, and I can be used andno letter can be used more than once?

Find the value of the permutation. -P(5, 1)

Solve the problem. -How many 3-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 if the first digitcannot be 0? Repeated digits are allowed.

Find the value of the permutation. -P(4, 0)

Solve the problem. -9 different books are to be arranged on a shelf. How many different arrangements are possible?

Find the value of the permutation. -P(9, 9)

Solve the problem. -How many different license plates can be made using 3 letters followed by 2 digits selected from the digits0 through 9, if letters and digits may be repeated?

Solve the problem. -How many 7-symbol codes can be formed using 4 different symbols? Repeated symbols are allowed.

Solve the problem. -The following data represent the marital status of females 18 years and older in a certain U.S. city. Determine the number of females 18 years old and older who are married or widowed.

Solve the problem. -A restaurant offers a choice of 4 salads, 7 main courses, and 2 desserts. How many possible 3-coursemeals are there?

Solve the problem. -How many arrangements of answers are possible in a multiple-choice test with 9 questions, each of whichhas 3 possible answers?

Solve the problem. -List all the ordered arrangements of 4 objects 1, 2, 3, and 4 choosing 3 at a time without repetition. What isP(4, 3) ?

Solve the problem. -A certain mathematics test consists of 20 questions. Goldie decides to answer the questions withoutreading them. In how many ways can Goldie fill in the answer sheet if the possible answers are true andfalse?

Solve the problem. -How many 2-letter codes can be formed using the letters A, B, C, D, E, F, G, H, and I. Repeated letters areallowed.

Solve the problem. -Lisa has 4 skirts, 8 blouses, and 2 jackets. How many 3-piece outfits can she put together assuming anypiece goes with any other?

Solve the problem. -List all the ordered arrangements of 6 objects a, b, c, d, e, and f choosing 2 at a time without repetition.What is P(6, 2) ?

Solve the problem. -A student must choose 1 of 4 mathematics electives, 1 of 6 science electives, and 1 of 9 programmingelectives. How many possible course selections are there?