Question 1

Solve the problem.
-A man has 10 shirts and 9 ties. How many different shirt and tie arrangements can he wear?

Accepted Answer

The number of different shirt and tie arrangements is the product of the number of shirts and the number of ties: 10 shirts * 9 ties = 90 arrangements.

Question 2

Find the value of the permutation.
-P(9, 1)

Accepted Answer

The value of the permutation P(9, 1) is calculated as 9! / (9-1)!, which simplifies to 9.

Question 3

Solve the problem.
-How many arrangements of answers are possible in a multiple-choice test with 7 questions, each of which has 5 possible answers?

Accepted Answer

For each of the 7 questions, there are 5 possible answers, so the total number of arrangements is $5^7 = 78,125$.

Question 4

Solve the problem.
-The following data represent the marital status of females 18 years and older in a certain U.S. city. $$\begin{array} { l | c } 
\text { Marital Status } & \text { Number (in thousands) } \
\hline \text { Married } & 308 \
\text { Widowed } & 58 \
\text { Divorced } & 65 \
\text { Never married } & 114
\end{array}$$ 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 5

Solve the problem.
-Among a group of 80 investors, 19 owned shares of Stock A, 29 owned shares of Stock B, 40 owned shares of Stock C, 12 owned shares of both Stock A and Stock B, 9 owned shares of Stock A and Stock C, 13 owned shares
Of Stock B and Stock C, and 7 owned shares of all three. How many investors did not have shares in any of the
Three? How many owned shares of either Stock A or Stock C but not Stock B?

Accepted Answer

The answer of Solve the problem.
-Among a group of 80...

Question 6

Solve the problem.
-Lisa has 4 skirts, 10 blouses, and 4 jackets. How many 3-piece outfits can she put together assuming any piece goes with any other?

Accepted Answer

Lisa can create outfits by combining 1 skirt, 1 blouse, and 1 jacket. The number of possible outfits is the product of the choices for each piece: 4 skirts * 10 blouses * 4 jackets = 160 possible outfits.

Question 7

Solve the problem.
-How many different license plates can be made using 3 letters followed by 3 digits selected from the digits 0 through 9, if letters and digits may be repeated?

Accepted Answer

For the three letters, since there are 26 letters in the alphabet and letters can be repeated, there are $26 \times 26 \times 26 = 17,576$ possibilities. For the three digits, since digits can range from 0 to 9 and can be repeated, there are $10 \times 10 \times 10 = 1,000$ possibilities. Multiplying these together gives $17,576 \times 1,000 = 17,576,000$ different license plates.

Question 8

Solve the problem.
-A student must choose 1 of 5 mathematics electives, 1 of 7 science electives, and 1 of 7 programming electives. How many possible course selections are there?

Accepted Answer

The answer of Solve the problem.
-A student must choose 1...

Question 9

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

The number of permutations of 6 objects taken 2 at a time is given by P(6, 2) = 6! / (6-2)! = 30. The list includes all possible ordered pairs without repetition.

Question 10

Solve the problem.
-A survey of 2,504 credit card users indicated that 1,167 had bought books online, 1,226 had bought music online, 431 had bought pet supplies online, 128 had bought both books and music, 227 had bought both books and pet
Supplies, 198 had bought both music and pet supplies, and 99 had bought all three. How many credit card users
Did not buy any of the three? How many bought either books or pet supplies but not music?

Accepted Answer

1,144 did not buy any of the three. 134 bought either books or pet supplies but not music.

Question 11

Solve the problem.
-In a survey of 189 vacationers in a popular beach resort town, 73 indicated they would consider buying a home there, 73 would consider buying a beach villa, 59 would consider buying a lot, 24 would consider both a home
And a villa, 27 would consider both a home and a lot, 26 would consider both a villa and a lot, and 12 would
Consider all three. How many vacationers would not consider any of the three? How many would consider only
A home?

Accepted Answer

To solve this problem, we use the principle of inclusion-exclusion to avoid double-counting those who are interested in more than one option. The total number of vacationers interested in at least one option is calculated as follows:- Total interested in a home, a villa, or a lot = 73 (home) + 73 (villa) + 59 (lot)- Subtract those interested in both options to avoid double counting = -24 (home and villa) - 27 (home and lot) - 26 (villa and lot)- Add back those interested in all three options, as they were subtracted twice in the previous step = +12This gives us: $73 + 73 + 59 - 24 - 27 - 26 + 12 = 140$Therefore, the number of vacationers not interested in any of the three options is the total number of vacationers minus those interested in at least one option: $189 - 140 = 49$.To find how many would consider only a home, we take the total number interested in a home and subtract those interested in a home and at least one other option:- Total interested in a home = 73- Subtract those interested in a home and at least one other option = -24 (home and villa) - 27 (home and lot) + 12 (since those interested in all three were subtracted twice)This gives us: $73 - 24 - 27 + 12 = 34$.Thus, 49 vacationers would not consider any of the three options, and 34 would consider only a home.

Question 12

Find the value of the permutation.
-P(11, 4)

Accepted Answer

The value of the permutation P(11, 4) is calculated using the formula P(n, k) = n! / (n-k)!, where n is the total number of items, and k is the number of items to be arranged. Here, n=11 and k=4, so P(11, 4) = 11! / (11-4)! = 11! / 7! = (11×10×9×8×7!)/7! = 11×10×9×8 = 7,920.

Question 13

Solve the problem.
-A restaurant offers a choice of 5 salads, 7 main courses, and 3 desserts. How many possible 3-course meals are there?

Accepted Answer

The answer of Solve the problem.
-A restaurant offers a choice...

Question 14

Find the value of the permutation.
-P(3, 3)

Accepted Answer

The permutation P(3, 3) is calculated as 3! (3 factorial), which is 3 × 2 × 1 = 6.

Question 15

Solve the problem.
-How many 4-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 if the first digit cannot be 0? Repeated digits are allowed.

Accepted Answer

For a 4-digit number, the first digit can be any of the 9 options (1-9), as 0 is not allowed. The remaining three digits can be any of the 10 digits (0-9), allowing for repetition. Therefore, the total number of possible 4-digit numbers is 9 * 10 * 10 * 10 = 9,000.

Question 16

Solve the problem.
-?

Accepted Answer

P(4, 3) represents the number of permutations of 4 items taken 3 at a time, which equals 4! / (4-3)! = 24. Choice A lists 24 unique permutations of the numbers 1, 2, 3, and 4 taken three at a time, matching the formula's result.

Question 17

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

Accepted Answer

The value of any permutation where the number of items to choose is 0 (P(n, 0)) is always 1, because there is exactly one way to choose nothing out of a set, regardless of the size of the set.

Question 18

Solve the problem.
-A certain mathematics test consists of 20 questions. Goldie decides to answer the questions without reading them. In how many ways can Goldie fill in the answer sheet if the possible answers are true and false?

Accepted Answer

For each question, Goldie has 2 choices (true or false). Since there are 20 questions, the total number of ways to fill in the answer sheet is $2^{20} = 1,048,576$.

Question 19

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 are allowed.

Accepted Answer

Since there are 9 letters to choose from for each of the 2 positions in the code, and repetition is allowed, the total number of possible codes is 9 * 9 = 81.

Question 20

Solve the problem.
-How many 6-symbol codes can be formed using 4 different symbols? Repeated symbols are allowed.

Accepted Answer

Since repetition is allowed and there are 4 different symbols to choose from for each of the 6 positions in the code, the total number of possible codes is $4^6 = 4,096$.

Quiz 12: Counting and Probability

Solve the problem. -A man has 10 shirts and 9 ties. How many different shirt and tie arrangements can he wear?

Find the value of the permutation. -P(9, 1)

Solve the problem. -How many arrangements of answers are possible in a multiple-choice test with 7 questions, each of which has 5 possible answers?

Solve the problem. -Lisa has 4 skirts, 10 blouses, and 4 jackets. How many 3-piece outfits can she put together assuming any piece goes with any other?

Solve the problem. -How many different license plates can be made using 3 letters followed by 3 digits selected from the digits 0 through 9, if letters and digits may be repeated?

Solve the problem. -A student must choose 1 of 5 mathematics electives, 1 of 7 science electives, and 1 of 7 programming electives. How many possible course selections are there?

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) ?

Find the value of the permutation. -P(11, 4)

Solve the problem. -A restaurant offers a choice of 5 salads, 7 main courses, and 3 desserts. How many possible 3-course meals are there?

Find the value of the permutation. -P(3, 3)

Solve the problem. -How many 4-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 if the first digit cannot be 0? Repeated digits are allowed.

Solve the problem. -?

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

Solve the problem. -A certain mathematics test consists of 20 questions. Goldie decides to answer the questions without reading them. In how many ways can Goldie fill in the answer sheet if the possible answers are true and false?

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 are allowed.

Solve the problem. -How many 6-symbol codes can be formed using 4 different symbols? Repeated symbols are allowed.