Question 1

Solve the problem.
-In a game of musical chairs, 9 children will sit in 8 chairs arranged in a row (one child will not find a chair). In how many ways can 8 of the children find seats?

Accepted Answer

The answer of Solve the problem.
-In a game of musical...

Question 2

Solve the problem.
-In how many ways can 2 letters be chosen from the set $\{A, B, C, D, E, F\}$ if order is important and no repeats are allowed?

Accepted Answer

Since order is important and no repeats are allowed, we use permutations. The number of ways to choose 2 letters from 6 is given by 6P2 = 6! / (6-2)! = 6 * 5 = 30.

Question 3

Solve the problem.
-A classical music concert is to consist of 5 cello pieces, 3 choral works, and 2 pieces for piano. In how many ways can the program be arranged if a piano piece must come first?

Accepted Answer

There are 10 pieces in total. If a piano piece must come first, there are 2 choices for the first piece. The remaining 9 pieces can be arranged in 9! ways. So, the total number of arrangements is $2 \times 9! = 725,760$.

Question 4

Solve the problem.
-In a lottery, players must pick 6 different numbers from 1 to 22 . How many ways are there to choose the 6 numbers if order is important?

Accepted Answer

When order is important, we use permutations. The number of permutations of 6 numbers from a set of 22 is calculated as $ \frac{22!}{(22-6)!} = \frac{22!}{16!} = 53,721,360 $.

Question 5

Solve the problem.
-From 10 names on a ballot, 3 will be elected to a political party committee. In how many ways can the committee of 3 be formed if each person will have a different responsibility?

Accepted Answer

The answer of Solve the problem.
-From 10 names on a...

Question 6

Solve the problem.
-The library is to be given 8 books as a gift. The books will be selected from a list of 20 titles. If each book selected must have a different title, how many possible selections are there?

Accepted Answer

The number of ways to select 8 books from 20 is given by the combination formula C(20, 8) = 20! / (8! * (20-8)!) = 125,970.

Question 7

Solve the problem.
-In a certain lottery, 4 numbers between 1 and 10 inclusive are drawn. These are the winning numbers. How many different selections are possible? Assume that the order in which the numbers are drawn is not important.

Accepted Answer

The problem is a combination problem where we need to choose 4 numbers out of 10 without regard to order. The number of combinations is calculated using the formula C(n, k) = n! / (k!(n-k)!), where n is the total number of items to choose from, and k is the number of items to choose. Here, n = 10 and k = 4, so C(10, 4) = 10! / (4! * 6!) = 210.

Question 8

Solve the problem.
-How many ways can an IRS auditor select 6 of 12 tax returns for an audit?

Accepted Answer

The number of ways to choose 6 tax returns from 12 is given by the combination formula $ \binom{12}{6} = \frac{12!}{6!(12-6)!} = 924 $.

Question 9

Solve the problem.
-Five cards are drawn at random from an ordinary deck of 52 cards. In how many ways is it possible to draw all black cards?

Accepted Answer

The answer of Solve the problem.
-Five cards are drawn at...

Question 10

Solve the problem.
-Five cards are drawn at random from an ordinary deck of 52 cards. In how many ways is it possible to draw two red cards and three black cards?

Accepted Answer

To solve this, we use combinations. There are 26 red cards and 26 black cards in a deck. We need to choose 2 red cards from 26 and 3 black cards from 26. The number of ways to choose 2 red cards is $ \binom{26}{2} $ and the number of ways to choose 3 black cards is $ \binom{26}{3} $. The total number of ways is the product of these two combinations: $ \binom{26}{2} \times \binom{26}{3} = 325 \times 260 = 845,000 $.

Question 11

Solve the problem.
-Five cards are drawn at random from an ordinary deck of 52 cards. In how many ways is it possible to draw two red aces and two black jacks?

Accepted Answer

There are 2 red aces and 2 black jacks in the deck. The number of ways to choose 2 red aces is 1 (since there are exactly 2 red aces), and the number of ways to choose 2 black jacks is also 1. The fifth card can be any of the remaining 48 cards (52 - 4 aces and jacks). Therefore, the total number of ways is 1 * 1 * 48 = 48 ways.

Question 12

Solve the problem.
-From a group of 10 women and 15 men, a researcher wants to randomly select 7 women and 7 men for a study. In how many ways can the study group be selected?

Accepted Answer

The answer of Solve the problem.
-From a group of 10...

Question 13

Solve the problem.
-Three student representatives are to be chosen from a group of five students: Andrew, Brenda, Chad, Dorothy, and Eric. In how many different ways can the representatives be chosen if two must be male and one female?

Accepted Answer

There are three males (Andrew, Chad, Eric) and two females (Brenda, Dorothy). We choose 2 males from 3 (3C2 = 3 ways) and 1 female from 2 (2C1 = 2 ways). Multiply the combinations: 3 * 2 = 6 ways.

Question 14

Solve the problem.
-Three student representatives, a president, a secretary, and a treasurer, are to be chosen from a group of five students: Andrew, Brenda, Chad, Dorothy, and Eric. In how many different ways can the representatives be chosen if the president must be a woman and the secretary and treasurer must be men?

Accepted Answer

There are two women (Brenda and Dorothy) who can be chosen as president. For the secretary and treasurer, we choose two out of the three men (Andrew, Chad, Eric), which can be done in 3 ways (since the order matters). Thus, there are 2 * 3 = 6 ways for each woman, totaling 12 ways.

Question 15

Solve the problem.
-Two student representatives, a treasurer and a secretary, are to be chosen from a group of five students: Andrew, Brenda, Chad, Dorothy, and Eric. In how many different ways can the representatives be chosen if the two must not be the same sex?

Accepted Answer

The answer of Solve the problem.
-Two student representatives, a treasurer...

Question 16

Solve the problem.
-Awards are to be presented to seven people: Jeff, Karen, Lyle, Maria, Norm, Olivia, and Paul. How many different orders are possible for the awards if Maria and Olivia must be next to each other?

Accepted Answer

The answer of Solve the problem.
-Awards are to be presented...

Question 17

Solve the problem.
-Awards are to be presented to seven people: Jeff, Karen, Lyle, Maria, Norm, Olivia, and Paul. How many different orders are possible for the awards if Karen is to receive the first award and Lyle the last?

Accepted Answer

Since Karen and Lyle's positions are fixed, we only need to arrange the remaining five people (Jeff, Maria, Norm, Olivia, and Paul), which can be done in 5! = 120 ways.

Question 18

Solve the problem.
-Awards are to be presented to seven people: Jeff, Karen, Lyle, Maria, Norm, Olivia, and Paul. How many different orders are possible for the awards if the men are to receive their awards first, and then the women?

Accepted Answer

The answer of Solve the problem.
-Awards are to be presented...

Question 19

Solve the problem.
-A bag contains 3 blue, 4 red, and 5 green marbles. Four marbles are drawn at random from the bag. How many different samples are possible which include exactly two red marbles?

Accepted Answer

To solve this, we use combinations. First, choose 2 red marbles from 4, which is C(4,2) = 6 ways. Then, choose 2 more marbles from the remaining 8 marbles (3 blue + 5 green), which is C(8,2) = 28 ways. Multiply these together: 6 * 28 = 168 different samples.

Question 20

Solve the problem.
-A bag contains 4 blue, 3 red, and 5 green marbles. Four marbles are drawn at random from the bag. How many different samples are possible which include at least one marble of each color?

Accepted Answer

The answer of Solve the problem.
-A bag contains 4 blue,...

Quiz 9: Counting, Probability Distributions, and Further Topics in Probability

Solve the problem. -In a game of musical chairs, 9 children will sit in 8 chairs arranged in a row (one child will not find a chair) . In how many ways can 8 of the children find seats?

Solve the problem. -In how many ways can 2 letters be chosen from the set $\{A, B, C, D, E, F\}$ if order is important and no repeats are allowed?

Solve the problem. -A classical music concert is to consist of 5 cello pieces, 3 choral works, and 2 pieces for piano. In how many ways can the program be arranged if a piano piece must come first?

Solve the problem. -In a lottery, players must pick 6 different numbers from 1 to 22 . How many ways are there to choose the 6 numbers if order is important?

Solve the problem. -From 10 names on a ballot, 3 will be elected to a political party committee. In how many ways can the committee of 3 be formed if each person will have a different responsibility?

Solve the problem. -The library is to be given 8 books as a gift. The books will be selected from a list of 20 titles. If each book selected must have a different title, how many possible selections are there?

Solve the problem. -In a certain lottery, 4 numbers between 1 and 10 inclusive are drawn. These are the winning numbers. How many different selections are possible? Assume that the order in which the numbers are drawn is not important.

Solve the problem. -How many ways can an IRS auditor select 6 of 12 tax returns for an audit?

Solve the problem. -Five cards are drawn at random from an ordinary deck of 52 cards. In how many ways is it possible to draw all black cards?

Solve the problem. -Five cards are drawn at random from an ordinary deck of 52 cards. In how many ways is it possible to draw two red cards and three black cards?

Solve the problem. -Five cards are drawn at random from an ordinary deck of 52 cards. In how many ways is it possible to draw two red aces and two black jacks?

Solve the problem. -From a group of 10 women and 15 men, a researcher wants to randomly select 7 women and 7 men for a study. In how many ways can the study group be selected?

Solve the problem. -Three student representatives are to be chosen from a group of five students: Andrew, Brenda, Chad, Dorothy, and Eric. In how many different ways can the representatives be chosen if two must be male and one female?

Solve the problem. -Two student representatives, a treasurer and a secretary, are to be chosen from a group of five students: Andrew, Brenda, Chad, Dorothy, and Eric. In how many different ways can the representatives be chosen if the two must not be the same sex?

Solve the problem. -Awards are to be presented to seven people: Jeff, Karen, Lyle, Maria, Norm, Olivia, and Paul. How many different orders are possible for the awards if Maria and Olivia must be next to each other?

Solve the problem. -Awards are to be presented to seven people: Jeff, Karen, Lyle, Maria, Norm, Olivia, and Paul. How many different orders are possible for the awards if Karen is to receive the first award and Lyle the last?

Solve the problem. -Awards are to be presented to seven people: Jeff, Karen, Lyle, Maria, Norm, Olivia, and Paul. How many different orders are possible for the awards if the men are to receive their awards first, and then the women?

Solve the problem. -A bag contains 3 blue, 4 red, and 5 green marbles. Four marbles are drawn at random from the bag. How many different samples are possible which include exactly two red marbles?

Solve the problem. -A bag contains 4 blue, 3 red, and 5 green marbles. Four marbles are drawn at random from the bag. How many different samples are possible which include at least one marble of each color?