Question 1

Solve the problem.
-How many arrangements of answers are possible in a multiple-choice test with 7 questions, each of which has 3 possible answers?

Accepted Answer

For each of the 7 questions, there are 3 possible answers. Therefore, the total number of arrangements is $3^7 = 2187$.

Question 2

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 options (true or false). Since there are 20 questions, the total number of ways to answer is $2^{20} = 1,048,576$.

Question 3

Find the value of the permutation.
-P(9, 0)

Accepted Answer

The value of any permutation P(n, 0) is always 1, because there is exactly one way to arrange zero objects out of a set of n objects (which is to not select any object at all).

Question 4

Solve the problem.
-How many different license plates can be made using 3 letters followed by 2 digits selected from the digits 0 through 9, if neither letters nor digits may be repeated?

Accepted Answer

There are 26 choices for each of the three letters and 10 choices for each of the two digits. So, the total number of different license plates is 26 * 26 * 26 * 10 * 10 = 1,404,000.

Question 5

Solve the problem.
-In how many ways can 6 people be lined up?

Accepted Answer

The number of ways to line up 6 people is calculated by finding the factorial of 6, which is 6! = 6*5*4*3*2*1 = 720.

Question 6

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 $4^6 = 4096$.

Question 7

Find the value of the permutation.
-P(7, 7)

Accepted Answer

The value of the permutation P(7, 7) is calculated as 7! (7 factorial), which equals 7*6*5*4*3*2*1 = 5040.

Question 8

Solve the problem.
-How many 5-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 5-digit number, the first digit can be any of the 9 options (1-9), and the remaining four can be any of the 10 digits (0-9), leading to 9 * 10^4 = 90,000 possibilities.

Question 9

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 repeated letters are allowed, for each of the 2 positions in the code, there are 9 possible letters that can be chosen (A, B, C, D, E, F, G, H, I). Therefore, the total number of possible codes is 9 * 9 = 81.

Question 10

Solve the problem.
-A student must choose 1 of 5 mathematics electives, 1 of 5 science electives, and 1 of 8 programming electives. How many possible course selections are there?

Accepted Answer

The total number of course selections is the product of the number of choices for each elective: 5 mathematics electives * 5 science electives * 8 programming electives = 200 course selections.

Question 11

Solve the problem.
-Lisa has 4 skirts, 9 blouses, and 3 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 * 9 blouses * 3 jackets = 36 * 3 = 108 possible outfits.

Question 12

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

Accepted Answer

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

Question 13

Find the value of the permutation.
-P(7, 1)

Accepted Answer

The value of the permutation P(7, 1) is calculated as 7! / (7-1)!, which simplifies to 7.

Question 14

Solve the problem.
-12 different books are to be arranged on a shelf. How many different arrangements are possible?

Accepted Answer

The number of different arrangements of 12 books is given by the factorial of 12, which is 12! = 479,001,600.

Question 15

Solve the problem.
-How many different license plates can be made using 2 letters followed by 2 digits selected from the digits 0 through 9, if letters and digits may be repeated?

Accepted Answer

For each of the 2 letters, there are 26 possibilities (since there are 26 letters in the English alphabet), and for each of the 2 digits, there are 10 possibilities (digits 0 through 9). Therefore, the total number of different license plates is 26 * 26 * 10 * 10 = 67,600.

Question 16

Find the value of the permutation.
-P(10, 3)

Accepted Answer

The answer of Find the value of the permutation.
-P(10, 3)
...

Question 17

Solve the problem.
-How many 2-digit numbers can be formed using the digits 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0? No digit can be used more than once.

Accepted Answer

For the first digit, you can choose any of the 9 options (1-9, 0 is not allowed because we can't have a leading zero in a 2-digit number). For the second digit, you can choose any of the remaining 9 digits (including 0 this time). So, the total number of combinations is 9 * 9 = 81. However, the correct calculation should be 9 options for the first digit and 10 options for the second digit (including 0), leading to 9 * 10 = 90 combinations.

Question 18

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

Accepted Answer

The answer of Solve the problem.
-List all the ordered arrangements...

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

The correct calculation for permutations of 6 objects taken 2 at a time without repetition is P(6, 2) = 6! / (6-2)! = 6*5 = 30, which matches the number of unique pairs listed in option C.

Question 20

Solve the problem.
-How many different 7-letter codes are there if only the letters A, B, C, D, E, F, G, H, and I can be used and no letter can be used more than once?

Accepted Answer

Since there are 9 letters to choose from and no letter can be used more than once, the number of different 7-letter codes can be calculated using permutations. The formula for permutations is nPr = n! / (n-r)!, where n is the total number of items to choose from, r is the number of items to choose, n! represents the factorial of n, and (n-r)! is the factorial of the difference between n and r. Here, n = 9 (letters A through I) and r = 7 (the length of the code). Therefore, the calculation is 9P7 = 9! / (9-7)! = 9! / 2! = 362,880 / 2 = 181,440.

Quiz 8: Counting and Probability

Solve the problem. -How many arrangements of answers are possible in a multiple-choice test with 7 questions, each of which has 3 possible answers?

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?

Find the value of the permutation. -P(9, 0)

Solve the problem. -How many different license plates can be made using 3 letters followed by 2 digits selected from the digits 0 through 9, if neither letters nor digits may be repeated?

Solve the problem. -In how many ways can 6 people be lined up?

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

Find the value of the permutation. -P(7, 7)

Solve the problem. -How many 5-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. -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. -A student must choose 1 of 5 mathematics electives, 1 of 5 science electives, and 1 of 8 programming electives. How many possible course selections are there?

Solve the problem. -Lisa has 4 skirts, 9 blouses, and 3 jackets. How many 3-piece outfits can she put together assuming any piece goes with any other?

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

Find the value of the permutation. -P(7, 1)

Solve the problem. -12 different books are to be arranged on a shelf. How many different arrangements are possible?

Solve the problem. -How many different license plates can be made using 2 letters followed by 2 digits selected from the digits 0 through 9, if letters and digits may be repeated?

Find the value of the permutation. -P(10, 3)

Solve the problem. -How many 2-digit numbers can be formed using the digits 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0? No digit can be used more than once.

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

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. -How many different 7-letter codes are there if only the letters A, B, C, D, E, F, G, H, and I can be used and no letter can be used more than once?