Question 1

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 a 2-digit number, the first digit can be any of the 9 options (1-9, since 0 cannot be the first digit), and the second digit can be any of the remaining 9 digits (including 0 this time). Thus, the total number of 2-digit numbers is 9 * 9 = 81.

Question 2

Solve the problem.
-How many different 8-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

This problem involves calculating permutations because we are arranging a set number of items (letters) and order matters. Since there are 9 letters to choose from and we want to arrange 8 of them without repetition, we use the permutation formula P(n, r) = n! / (n-r)!, where n is the total number of items to choose from, r is the number of items to arrange, and "!" denotes factorial (the product of all positive integers up to that number). Here, n = 9 and r = 8, so P(9, 8) = 9! / (9-8)! = 9! / 1! = 9! = 362,880. However, the correct answer is not listed directly, indicating a mistake in my explanation. The correct calculation should be for arranging 8 letters out of 9, which indeed is 9! / (9-8)! = 9! / 1! = 9! = 362,880, but the correct answer choice provided is D) 181,440, which suggests a misunderstanding in the calculation process. The correct approach is to calculate the permutations of 8 out of 9 letters, which is 9P8 = 9! / (9-8)! = 9! / 1! = 362,880, not 181,440. Therefore, there seems to be a discrepancy in the interpretation of the answer choices.

Question 3

Solve the problem.
-How many 5-card poker hands consisting of three 5's and two cards that are not 5's are possible in a 52-card deck?

Accepted Answer

The answer of Solve the problem.
-How many 5-card poker hands...

Question 4

Solve the problem.
-How many different license plates can be made using 3 letters followed by 4 digits selected from the digits 0 through 9, if digits may be repeated but letters may not be repeated?

Accepted Answer

The answer of Solve the problem.
-How many different license plates...

Question 5

Solve the problem.
-How many ways are there to choose a soccer team consisting of 3 forwards, 4 midfield players, and 3 defensive players, if the players are chosen from 6 forwards, 7 midfield players, and 5 defensive players?

Accepted Answer

To form the team, we choose 3 forwards from 6, 4 midfield players from 7, and 3 defensive players from 5. The number of ways to choose the players is calculated by multiplying the combinations: C(6,3) * C(7,4) * C(5,3) = 20 * 35 * 10 = 7,000.

Question 6

Solve the problem.
-6 different books are to be arranged on a shelf. How many different arrangements are possible?

Accepted Answer

The number of ways to arrange 6 different books on a shelf is given by 6! (6 factorial), which equals 720.

Question 7

Find the value of the combination.
-C(5, 1)

Accepted Answer

The combination formula is C(n, k) = n! / [k!(n-k)!]. For C(5, 1), it simplifies to 5! / [1!(5-1)!] = 5 / 1 = 5.

Question 8

Solve the problem.
-A group of 7 friends goes bowling. How many different possibilities are there for the order in which they play if the youngest person is to bowl first?

Accepted Answer

Since the youngest person is fixed to bowl first, we only need to arrange the remaining 6 friends. This can be done in 6! (6 factorial) ways, which is 720.

Question 9

Solve the problem.
-How many different 11-letter words (real or imaginary) can be formed from the letters in the word ENGINEERING?

Accepted Answer

The word ENGINEERING has 11 letters in total, with the following repetitions: E appears 3 times, N appears 3 times, G appears 2 times, and I appears 2 times. The formula for permutations of a multiset is n! / (n1! * n2! * ... * nk!), where n is the total number of items, and n1, n2, ..., nk are the counts of each repeated item. Applying this formula, we get 11! / (3! * 3! * 2! * 2!) = 277,200.

Question 10

Solve the problem.
-List all the combinations of 6 objects a, b, c, d, e, and f taken 2 at a time. What is C(6, 2)?

Accepted Answer

C(6, 2) calculates the number of combinations of 6 objects taken 2 at a time, which is calculated as 6! / (2!(6-2)!) = 15. This means each pair is counted only once, without regard to order, which matches option D.

Question 11

Solve the problem.
-An exam consists of 9 multiple-choice questions and 6 essay questions. If the student must answer 5 of the multiple-choice questions and 3 of the essay questions, in how many ways can the questions be chosen?

Accepted Answer

The number of ways to choose 5 out of 9 multiple-choice questions is calculated using combinations: 9C5 = 9! / (5!(9-5)!) = 126. The number of ways to choose 3 out of 6 essay questions is calculated using combinations: 6C3 = 6! / (3!(6-3)!) = 20. To find the total number of ways to choose the questions, multiply the two results: 126 * 20 = 2520.

Question 12

Solve the problem.
-List all the combinations of 4 objects 1, 2, 3, and 4 taken 3 at a time. What is C(4, 3)?

Accepted Answer

The combinations of 4 objects taken 3 at a time are 123, 124, 134, and 234. The number of combinations is given by C(4, 3) = 4.

Question 13

Solve the problem.
-A committee is to be formed consisting of 2 men and 3 women. If the committee members are to be chosen from 13 men and 9 women, how many different committees are possible?

Accepted Answer

The number of ways to choose 2 men out of 13 is $C(13, 2) = \frac{13!}{2!(13-2)!} = 78$. The number of ways to choose 3 women out of 9 is $C(9, 3) = \frac{9!}{3!(9-3)!} = 84$. The total number of different committees is the product of these two, which is $78 \times 84 = 6,552$.

Question 14

Solve the problem.
-Mary finds 8 fish at a pet store that she would like to buy, but she can afford only 5 of them. In how many ways can she make her selection? How many ways can she make her selection if he decides that one of the fish is a
Must?

Accepted Answer

Mary can select 5 out of 8 fish in $ \binom{8}{5} = 56 $ ways. If one fish is a must, she needs to choose 4 out of the remaining 7, which can be done in $ \binom{7}{4} = 35 $ ways.

Question 15

Solve the problem.
-In how many ways can 2 people each have different birth months?

Accepted Answer

The first person can have a birth month in any of the 12 months, and the second person can have a birth month in any of the remaining 11 months (since both must have different birth months). Therefore, the total number of ways is 12 * 11 = 132.

Question 16

Solve the problem.
-A hot dog stand sells hot dogs with cheese, relish, chili, tomato, onion, mustard, or ketchup. How many different hot dogs can be concocted using any 3 of the extras?

Accepted Answer

The answer of Solve the problem.
-A hot dog stand sells...

Question 17

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

Accepted Answer

The answer of Solve the problem.
-How many different license plates...

Question 18

Solve the problem.
-In how many ways can 12 people be lined up?

Accepted Answer

The number of ways to line up 12 people is calculated by finding the factorial of 12, which is 12! = 12 x 11 x 10 x ... x 1 = 479,001,600.

Question 19

Find the value of the combination.
-C(7, 7)

Accepted Answer

The combination formula is C(n, k) = n! / [k!(n-k)!]. For C(7, 7), it simplifies to 7! / [7!0!] = 1, since 0! = 1.

Question 20

Solve the problem.
-From 9 names on a ballot, a committee of 5 will be elected to attend a political national convention. How many different committees are possible?

Accepted Answer

The number of different committees can be calculated using the combination formula C(n, k) = n! / (k!(n-k)!), where n is the total number of names and k is the number of names to choose. Here, C(9, 5) = 9! / (5! * (9-5)!) = 126.

Quiz 12: Counting and Probability

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. -How many different 8-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?

Solve the problem. -How many 5-card poker hands consisting of three 5's and two cards that are not 5's are possible in a 52-card deck?

Solve the problem. -How many different license plates can be made using 3 letters followed by 4 digits selected from the digits 0 through 9, if digits may be repeated but letters may not be repeated?

Solve the problem. -How many ways are there to choose a soccer team consisting of 3 forwards, 4 midfield players, and 3 defensive players, if the players are chosen from 6 forwards, 7 midfield players, and 5 defensive players?

Solve the problem. -6 different books are to be arranged on a shelf. How many different arrangements are possible?

Find the value of the combination. -C(5, 1)

Solve the problem. -A group of 7 friends goes bowling. How many different possibilities are there for the order in which they play if the youngest person is to bowl first?

Solve the problem. -How many different 11-letter words (real or imaginary) can be formed from the letters in the word ENGINEERING?

Solve the problem. -List all the combinations of 6 objects a, b, c, d, e, and f taken 2 at a time. What is C(6, 2) ?

Solve the problem. -An exam consists of 9 multiple-choice questions and 6 essay questions. If the student must answer 5 of the multiple-choice questions and 3 of the essay questions, in how many ways can the questions be chosen?

Solve the problem. -List all the combinations of 4 objects 1, 2, 3, and 4 taken 3 at a time. What is C(4, 3) ?

Solve the problem. -A committee is to be formed consisting of 2 men and 3 women. If the committee members are to be chosen from 13 men and 9 women, how many different committees are possible?

Solve the problem. -Mary finds 8 fish at a pet store that she would like to buy, but she can afford only 5 of them. In how many ways can she make her selection? How many ways can she make her selection if he decides that one of the fish is a Must?

Solve the problem. -In how many ways can 2 people each have different birth months?

Solve the problem. -A hot dog stand sells hot dogs with cheese, relish, chili, tomato, onion, mustard, or ketchup. How many different hot dogs can be concocted using any 3 of the extras?

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

Solve the problem. -In how many ways can 12 people be lined up?

Find the value of the combination. -C(7, 7)

Solve the problem. -From 9 names on a ballot, a committee of 5 will be elected to attend a political national convention. How many different committees are possible?