Question 1

Solve the problem.
-A group of 12 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 11 people. The number of ways to arrange 11 people is 11! (11 factorial), which equals 39,916,800.

Question 2

An environmental organization has 20 members. Each member will be placed on exactly 1 of 4 teams. Each team will work on a different issue. The first team has 5 members, the second has 4, the third has 8, and the
Fourth has 3. In how many ways can these teams be formed?

Accepted Answer

The answer of An environmental organization has 20 members. Each...

Question 3

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 4

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 5 of the extras?

Accepted Answer

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

Question 5

Find the value of the combination.
-C(6, 6)

Accepted Answer

The combination formula is C(n, k) = n! / [k!(n-k)!]. For C(6, 6), it simplifies to 6! / [6!0!] = 1, since 0! = 1.

Question 6

In a probability model, which of the following numbers could be the probability of an outcome: $0,0.2 , - 0.01 , - \frac { 1 } { 3 } , \frac { 1 } { 2 } , \frac { 5 } { 4 } , 1,1.5$

Accepted Answer

Probabilities must be between 0 and 1, inclusive.

Question 7

Solve the problem.
-In how many ways can 7 people each have different birth months?

Accepted Answer

There are 12 months and we need to assign a different month to each of the 7 people. This is a permutation problem: 12P7 = 12! / (12-7)! = 3,991,680.

Question 8

Determine whether the following is a probability model.
-$\begin{array} { l | c } 
\text { Outcome } & \text { Probability } \
\hline \text { Red } & 0.21 \
\text { Blue } & 0.23 \
\text { Green } & 0.24 \
\text { White } & 0.32
\end{array}$

Accepted Answer

The sum of the probabilities is 1 (0.21 + 0.23 + 0.24 + 0.32 = 1.00), which is a requirement for a probability model.

Question 9

Solve the problem.
-Mary finds 10 fish at a pet store that she would like to buy, but she can afford only 4 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

The number of ways Mary can select 4 fish out of 10 is calculated using combinations since the order of selection does not matter. This is given by 10C4 = 10! / (4!(10-4)!) = 210 ways. If one fish is a must, then she only needs to choose 3 out of the remaining 9, which is 9C3 = 9! / (3!(9-3)!) = 84 ways.

Question 10

How many different vertical arrangements are there of 8 flags if 4 are white, 2 are blue, and 2 are red?

Accepted Answer

The total number of arrangements can be calculated using the formula for permutations of a multiset: $ \frac{n!}{n_1!n_2!n_3!...} $, where $ n $ is the total number of items, and $ n_1, n_2, n_3, ... $ are the counts of identical items. Here, $ n = 8 $ (total flags), $ n_1 = 4 $ (white flags), $ n_2 = 2 $ (blue flags), and $ n_3 = 2 $ (red flags). So, the total number of arrangements is $ \frac{8!}{4!2!2!} = 420 $.

Question 11

Solve the problem.
-An exam consists of 9 multiple-choice questions and 6 essay questions. If the student must answer 6 of the multiple-choice questions and 4 of the essay questions, in how many ways can the questions be chosen?

Accepted Answer

The number of ways to choose 6 multiple-choice questions out of 9 is calculated using combinations: 9C6 = 9! / (6!(9-6)!) = 84. The number of ways to choose 4 essay questions out of 6 is calculated using combinations: 6C4 = 6! / (4!(6-4)!) = 15. The total number of ways to choose the questions is the product of these two combinations: 84 * 15 = 1260.

Question 12

Solve the problem.
-A committee is to be formed consisting of 2 men and 5 women. If the committee members are to be chosen from 9 men and 12 women, how many different committees are possible?

Accepted Answer

The number of ways to choose 2 men from 9 is C(9,2) = 36, and the number of ways to choose 5 women from 12 is C(12,5) = 792. The total number of different committees is the product of these two combinations: 36 * 792 = 28,512.

Question 13

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 8 forwards, 10 midfield players, and 8 defensive players?

Accepted Answer

The number of ways to choose the soccer team can be calculated using combinations since the order in which the players are chosen does not matter. For forwards, there are $C(8,3) = \frac{8!}{3!(8-3)!} = 56$ ways. For midfield players, $C(10,4) = \frac{10!}{4!(10-4)!} = 210$ ways. For defensive players, $C(8,3) = \frac{8!}{3!(8-3)!} = 56$ ways. The total number of ways to choose the team is the product of these, which is $56 \times 210 \times 56 = 658,560$.

Question 14

Solve the problem.
-How many different license plates can be made using 4 letters followed by 2 digits selected from the digits 0 through 9, if digits may be repeated but letters may not be repeated?

Accepted Answer

For the letters, since repetition is not allowed, the first letter can be any of the 26 letters in the alphabet, the second can be any of the remaining 25, the third can be any of the remaining 24, and the fourth can be any of the remaining 23. This gives us 26 * 25 * 24 * 23 possible combinations for the letters. For the digits, since repetition is allowed, each of the two digits can be any of the 10 digits from 0 to 9, giving us 10 * 10 possibilities for the digits. Multiplying these together gives us the total number of different license plates possible: 26 * 25 * 24 * 23 * 10 * 10 = 35,880,000.

Question 15

How many different 11-letter words (real or imaginary) can be formed from the letters of the word MISSISSIPPI? Leave your answer in factorial form.

Accepted Answer

The answer of How many different 11-letter words (real or...

Question 16

Solve the problem.
-From 9 names on a ballot, a committee of 4 will be elected to attend a political national convention. How many different committees are possible?

Accepted Answer

The number of ways to choose 4 names out of 9 for a committee is calculated using the combination formula, which is C(n, k) = n! / [k!(n-k)!], where n is the total number of items, k is the number of items to choose, and "!" denotes factorial. Here, n=9 and k=4, so C(9, 4) = 9! / [4!(9-4)!] = 9! / (4!5!) = (9*8*7*6) / (4*3*2*1) = 126.

Question 17

Find the value of the combination.
-C(10, 6)

Accepted Answer

The value of the combination C(10, 6) is calculated using the formula C(n, k) = n! / [k!(n-k)!], where n = 10 and k = 6. Thus, C(10, 6) = 10! / [6!(10-6)!] = 10! / (6!4!) = (10*9*8*7)/(4*3*2*1) = 210.

Question 18

Solve the problem.
-How many different 10-letter words (real or imaginary) can be formed from the letters in the word IMMUNOLOGY?

Accepted Answer

The word IMMUNOLOGY has 10 letters with the following counts of identical letters: I-2, M-2, O-2. The formula for permutations of a set of objects where some objects are identical is n! / (n1! * n2! * ... * nk!), where n is the total number of objects, and n1, n2, ..., nk are the counts of each type of identical object. Here, n=10 (total letters), and we have three sets of identical letters (I, M, O) each appearing twice. Thus, the number of different 10-letter words that can be formed is 10! / (2! * 2! * 2!) = 907,200.

Question 19

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

The correct answer is A because C(6, 2) calculates the number of combinations of 6 items taken 2 at a time without regard to order, which equals 15.

Question 20

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...

Quiz 8: Counting and Probability

Solve the problem. -A group of 12 friends goes bowling. How many different possibilities are there for the order in which they play if the youngest person is to bowl first?

An environmental organization has 20 members. Each member will be placed on exactly 1 of 4 teams. Each team will work on a different issue. The first team has 5 members, the second has 4, the third has 8, and the Fourth has 3. In how many ways can these teams be formed?

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 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 5 of the extras?

Find the value of the combination. -C(6, 6)

In a probability model, which of the following numbers could be the probability of an outcome: $0,0.2 , - 0.01 , - \frac { 1 } { 3 } , \frac { 1 } { 2 } , \frac { 5 } { 4 } , 1,1.5$

Solve the problem. -In how many ways can 7 people each have different birth months?

Determine whether the following is a probability model. - $\begin{array} { l | c } \text { Outcome } & \text { Probability } \\\hline \text { Red } & 0.21 \\\text { Blue } & 0.23 \\\text { Green } & 0.24 \\\text { White } & 0.32\end{array}$

Solve the problem. -Mary finds 10 fish at a pet store that she would like to buy, but she can afford only 4 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?

How many different vertical arrangements are there of 8 flags if 4 are white, 2 are blue, and 2 are red?

Solve the problem. -An exam consists of 9 multiple-choice questions and 6 essay questions. If the student must answer 6 of the multiple-choice questions and 4 of the essay questions, in how many ways can the questions be chosen?

Solve the problem. -A committee is to be formed consisting of 2 men and 5 women. If the committee members are to be chosen from 9 men and 12 women, how many different committees are possible?

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 8 forwards, 10 midfield players, and 8 defensive players?

Solve the problem. -How many different license plates can be made using 4 letters followed by 2 digits selected from the digits 0 through 9, if digits may be repeated but letters may not be repeated?

How many different 11-letter words (real or imaginary) can be formed from the letters of the word MISSISSIPPI? Leave your answer in factorial form.

Solve the problem. -From 9 names on a ballot, a committee of 4 will be elected to attend a political national convention. How many different committees are possible?

Find the value of the combination. -C(10, 6)

Solve the problem. -How many different 10-letter words (real or imaginary) can be formed from the letters in the word IMMUNOLOGY?

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. -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?

Quiz 8: Counting and Probability

Solve the problem. -A group of 12 friends goes bowling. How many different possibilities are there for the order in which they play if the youngest person is to bowl first?

An environmental organization has 20 members. Each member will be placed on exactly 1 of 4 teams. Each team will work on a different issue. The first team has 5 members, the second has 4, the third has 8, and the Fourth has 3. In how many ways can these teams be formed?

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 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 5 of the extras?

Find the value of the combination. -C(6, 6)

In a probability model, which of the following numbers could be the probability of an outcome: 0,0.2,−0.01,−13,12,54,1,1.50,0.2 , - 0.01 , - \frac { 1 } { 3 } , \frac { 1 } { 2 } , \frac { 5 } { 4 } , 1,1.50,0.2,−0.01,−31​,21​,45​,1,1.5

Solve the problem. -In how many ways can 7 people each have different birth months?

Solve the problem. -Mary finds 10 fish at a pet store that she would like to buy, but she can afford only 4 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?

How many different vertical arrangements are there of 8 flags if 4 are white, 2 are blue, and 2 are red?

Solve the problem. -An exam consists of 9 multiple-choice questions and 6 essay questions. If the student must answer 6 of the multiple-choice questions and 4 of the essay questions, in how many ways can the questions be chosen?

Solve the problem. -A committee is to be formed consisting of 2 men and 5 women. If the committee members are to be chosen from 9 men and 12 women, how many different committees are possible?

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 8 forwards, 10 midfield players, and 8 defensive players?

Solve the problem. -How many different license plates can be made using 4 letters followed by 2 digits selected from the digits 0 through 9, if digits may be repeated but letters may not be repeated?

How many different 11-letter words (real or imaginary) can be formed from the letters of the word MISSISSIPPI? Leave your answer in factorial form.

Solve the problem. -From 9 names on a ballot, a committee of 4 will be elected to attend a political national convention. How many different committees are possible?

Find the value of the combination. -C(10, 6)

Solve the problem. -How many different 10-letter words (real or imaginary) can be formed from the letters in the word IMMUNOLOGY?

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. -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?

In a probability model, which of the following numbers could be the probability of an outcome: $0,0.2 , - 0.01 , - \frac { 1 } { 3 } , \frac { 1 } { 2 } , \frac { 5 } { 4 } , 1,1.5$