Question 1

Which of the following is false?
A) $13 !=13 \cdot 12 \cdot 11 \cdot 10 !$
B) $8^{P_{2}}=28$
C) $\left(\begin{array}{l}5 \ 2\end{array}\right)=10$
D) $\left(\begin{array}{c}15 \ 4\end{array}\right)=\frac{15 \cdot 14 \cdot 13 \cdot 12}{4 \cdot 3 \cdot 2}$

Accepted Answer

- A) is true because $13!$ means $13 \times 12 \times 11 \times ... \times 1$, so the expression correctly simplifies to $13 \times 12 \times 11 \times 10!$.- B) is false because $8^{P_{2}}$ refers to the permutation of 8 items taken 2 at a time, which is $8 \times 7 = 56$, not 28.- C) is true because $\left(\begin{array}{l}5 \ 2\end{array}\right)$ represents the combination of 5 items taken 2 at a time, which is $\frac{5 \times 4}{2 \times 1} = 10$.- D) is true because $\left(\begin{array}{c}15 \ 4\end{array}\right)$ represents the combination of 15 items taken 4 at a time, and the given formula correctly calculates this value.

Question 2

A multiple choice test has 10 questions with 4 choices for each question. The number of ways a student can answer all 10 questions is
A) $10 \cdot 4$
B) 410
C) $10^{4}$
D) $10+4$

Accepted Answer

For each of the 10 questions, there are 4 choices, so the total number of ways to answer all questions is $4^{10}$.

Question 3

Two restaurants are to be built at locations selected from a choice of 11 locations. The number of ways selecting the locations for the two restaurants is
A) $\frac{11 !}{2}$
B) $11 \cdot 11$
C) $\frac{11 \cdot 10}{2}$
D) $11 \cdot 10$

Accepted Answer

The correct answer is found by using the combination formula for selecting 2 locations out of 11, which is $C(n, k) = \frac{n!}{k!(n-k)!}$. Here, $n=11$ and $k=2$, so it simplifies to $\frac{11 \cdot 10}{2}$. This accounts for the fact that the order of selection does not matter (selecting location A then B is the same as selecting B then A).

Question 4

In an 8-horse race, the number of ways that 3 horses can finish first, second, and third (order matters) is given by
A) $8 \cdot 3$.
B) $8^{3}$.
C) $8 \cdot 7 \cdot 6$
D) $\frac{8 \cdot 7 \cdot 6}{3 \cdot 2 \cdot 1}$

Accepted Answer

This is a permutation problem where order matters. The first place can be taken by any of the 8 horses, the second place by any of the remaining 7, and the third by any of the remaining 6, giving $8 \cdot 7 \cdot 6$ ways.

Question 5

The probability of selecting a defective component from 20 components, 3 of which are defective, is $\frac{3}{20}$. This is an example of
A) the classical probability concept.
B) the frequency interpretation of probability.
C) the law of large numbers.
D) subjective probability.

Accepted Answer

The classical probability concept is used here, where the probability is calculated as the ratio of the number of favorable outcomes (selecting a defective component) to the total number of outcomes (total components).

Question 6

A company employs 40 skilled and 30 unskilled workers. Twenty -five of the skilled workers and 17 of the unskilled are college graduates. If an employee of the company is selected at random, the probability that the person picked will not be a college graduate is
A) $\frac{2}{5}$.
B) $\frac{25}{70}$.
C) $\frac{12}{28}$.
D) $\frac{17}{70}$.

Accepted Answer

The total number of workers is 40 skilled + 30 unskilled = 70 workers. The number of college graduates is 25 skilled + 17 unskilled = 42. Therefore, the number of workers who are not college graduates is 70 - 42 = 28. The probability that a randomly selected employee is not a college graduate is 28 out of 70, which simplifies to $\frac{2}{5}$.

Question 7

Which of the following is not true of the symbol $\left(\begin{array}{c}11 \ 3\end{array}\right)$ ?
A) It is used when order does not matter.
B) Its value is less than $11 P_{3}$.
C) It is the number of 3 letter words that can be formed from 11 letters.
D) It is the number of 3 person committees that can be formed from a group of 11 people.

Accepted Answer

- A) is true because the symbol represents a combination, which is used in situations where order does not matter.- B) is true because combinations are generally less than permutations for the same values since permutations consider order, making them larger.- D) is true because this is a classic example of when combinations are used: selecting a subset (like a committee) from a larger set without regard to the order of selection.- C) is not true because the symbol represents combinations, not permutations, and the formation of words would consider the order, making permutations the correct approach, not combinations.

Question 8

In the past eight Friday nights, a student has gone to the same rock music club on five occasions. The claim that the probability is $\frac{5}{8}$ that the student will attend the same club next Friday is based on
A) the classical probability concept.
B) the frequency interpretation of probability.
C) the law of large numbers.
D) subjective probability.

Accepted Answer

The frequency interpretation of probability uses the past frequency of an event occurring to predict the likelihood of that event happening in the future. Since the student attended the club 5 out of 8 times in the past, the claim uses this historical frequency to estimate the probability of future attendance.

Question 9

A student has time to see four films on his vacation. If he is selecting the movies from a total of nine choices, how many ways can he choose the films?
A) $9 \cdot 4$
B) $9 \cdot 8 \cdot 7 \cdot 6$
C) $\frac{9 !}{4 !}$
D) $\frac{9 \cdot 8 \cdot 7 \cdot 6}{4 \cdot 3 \cdot 2 \cdot 1}$

Accepted Answer

The correct way to calculate the number of ways to choose 4 films out of 9 is to use the combination formula, which is $\frac{n!}{r!(n-r)!}$, where $n$ is the total number of items (9 films) and $r$ is the number of items to choose (4 films). This simplifies to $\frac{9!}{4!(9-4)!}$, which further simplifies to $\frac{9 \cdot 8 \cdot 7 \cdot 6}{4 \cdot 3 \cdot 2 \cdot 1}$, matching option D.

Question 10

A student has seven items of living room furniture and room for five items in his living room. Assuming that any of the seven items can be placed in any of the five locations, in how many ways can the student arrange his living room with the furniture?
A) $\left(\begin{array}{l}7 \ 5\end{array}\right)$
B) $7 \cdot 6$
C) $7 \cdot 6 \cdot 5 \cdot 4 \cdot 3$
D) $7 \cdot 5$

Accepted Answer

The correct answer is found by using the permutation formula for selecting 5 items out of 7, which is $7 \cdot 6 \cdot 5 \cdot 4 \cdot 3$, because the order in which the furniture is arranged matters.

Question 11

A student can play 0,1 , or 2 hours of racquetball on any given night. Construct a tree diagram to determine the number of ways that in three nights he can play for a total of four hours.

Accepted Answer

The answer of A student can play 0,1 , or...

Question 12

A student can play 0, 1, or 2 hours of racquetball on any given night. Construct a tree diagram to determine the number of ways that in three nights he can play for a total of five hours.

Accepted Answer

The answer of A student can play 0, 1, or...

Question 13

Give the number of ways a student can mark her answers to a multiple choice test if there are eight questions with three choices each.

Accepted Answer

The answer of Give the number of ways a student...

Question 14

Give the number of ways a student can mark her answers to a multiple choice test if there are three questions with eight choices each

Accepted Answer

The answer of Give the number of ways a student...

Question 15

Give the number of ways a student can mark her answers to a multiple choice test if there are 10 questions with four choices each for the first three questions and five choices each for the next seven questions.

Accepted Answer

The answer of Give the number of ways a student...

Question 16

There are six baseball teams in the American League's Eastern division. In how many ways can:}
-three teams finish first, second, and third?

Accepted Answer

The answer of There are six baseball teams in the...

Question 17

There are six baseball teams in the American League's Eastern division. In how many ways can:}
-all six teams finish first through sixth?

Accepted Answer

The answer of There are six baseball teams in the...

Question 18

A new campus project requires a committee of 5 students which will include 2 seniors and 3 juniors. There are 6 available seniors and 8 available juniors.
-In how many ways can the juniors be chosen for the committee?

Accepted Answer

The answer of A new campus project requires a committee...

Question 19

A new campus project requires a committee of 5 students which will include 2 seniors and 3 juniors. There are 6 available seniors and 8 available juniors.
-In how many ways can the required committee be chosen?

Accepted Answer

The answer of A new campus project requires a committee...

Question 20

A new campus project requires a committee of 5 students which will include 2 seniors and 3 juniors. There are 6 available seniors and 8 available juniors.
-In how many ways can the committee be chosen from the available students if there are no restrictions on the number of juniors or seniors on the committee?

Accepted Answer

The answer of A new campus project requires a committee...

Quiz 5: Possibilities and Probabilities

Which of the following is false?

A multiple choice test has 10 questions with 4 choices for each question. The number of ways a student can answer all 10 questions is

Two restaurants are to be built at locations selected from a choice of 11 locations. The number of ways selecting the locations for the two restaurants is

In an 8-horse race, the number of ways that 3 horses can finish first, second, and third (order matters) is given by

The probability of selecting a defective component from 20 components, 3 of which are defective, is $\frac{3}{20}$ . This is an example of

A company employs 40 skilled and 30 unskilled workers. Twenty -five of the skilled workers and 17 of the unskilled are college graduates. If an employee of the company is selected at random, the probability that the person picked will not be a college graduate is

Which of the following is not true of the symbol $\left(\begin{array}{c}11 \\ 3\end{array}\right)$ ?

In the past eight Friday nights, a student has gone to the same rock music club on five occasions. The claim that the probability is $\frac{5}{8}$ that the student will attend the same club next Friday is based on

A student has time to see four films on his vacation. If he is selecting the movies from a total of nine choices, how many ways can he choose the films?

A student has seven items of living room furniture and room for five items in his living room. Assuming that any of the seven items can be placed in any of the five locations, in how many ways can the student arrange his living room with the furniture?

A student can play 0,1 , or 2 hours of racquetball on any given night. Construct a tree diagram to determine the number of ways that in three nights he can play for a total of four hours.

A student can play 0, 1, or 2 hours of racquetball on any given night. Construct a tree diagram to determine the number of ways that in three nights he can play for a total of five hours.

Give the number of ways a student can mark her answers to a multiple choice test if there are eight questions with three choices each.

Give the number of ways a student can mark her answers to a multiple choice test if there are three questions with eight choices each

Give the number of ways a student can mark her answers to a multiple choice test if there are 10 questions with four choices each for the first three questions and five choices each for the next seven questions.

There are six baseball teams in the American League's Eastern division. In how many ways can:} -three teams finish first, second, and third?

There are six baseball teams in the American League's Eastern division. In how many ways can:} -all six teams finish first through sixth?

A new campus project requires a committee of 5 students which will include 2 seniors and 3 juniors. There are 6 available seniors and 8 available juniors. -In how many ways can the juniors be chosen for the committee?

A new campus project requires a committee of 5 students which will include 2 seniors and 3 juniors. There are 6 available seniors and 8 available juniors. -In how many ways can the required committee be chosen?

Quiz 5: Possibilities and Probabilities

Which of the following is false?

A multiple choice test has 10 questions with 4 choices for each question. The number of ways a student can answer all 10 questions is

Two restaurants are to be built at locations selected from a choice of 11 locations. The number of ways selecting the locations for the two restaurants is

In an 8-horse race, the number of ways that 3 horses can finish first, second, and third (order matters) is given by

The probability of selecting a defective component from 20 components, 3 of which are defective, is 320\frac{3}{20}203​ . This is an example of

A company employs 40 skilled and 30 unskilled workers. Twenty -five of the skilled workers and 17 of the unskilled are college graduates. If an employee of the company is selected at random, the probability that the person picked will not be a college graduate is

Which of the following is not true of the symbol (113) \left(\begin{array}{c}11 \\ 3\end{array}\right) (113​) ?

In the past eight Friday nights, a student has gone to the same rock music club on five occasions. The claim that the probability is 58\frac{5}{8}85​ that the student will attend the same club next Friday is based on

A student has time to see four films on his vacation. If he is selecting the movies from a total of nine choices, how many ways can he choose the films?

A student has seven items of living room furniture and room for five items in his living room. Assuming that any of the seven items can be placed in any of the five locations, in how many ways can the student arrange his living room with the furniture?

A student can play 0,1 , or 2 hours of racquetball on any given night. Construct a tree diagram to determine the number of ways that in three nights he can play for a total of four hours.

A student can play 0, 1, or 2 hours of racquetball on any given night. Construct a tree diagram to determine the number of ways that in three nights he can play for a total of five hours.

Give the number of ways a student can mark her answers to a multiple choice test if there are eight questions with three choices each.

Give the number of ways a student can mark her answers to a multiple choice test if there are three questions with eight choices each

Give the number of ways a student can mark her answers to a multiple choice test if there are 10 questions with four choices each for the first three questions and five choices each for the next seven questions.

There are six baseball teams in the American League's Eastern division. In how many ways can:} -three teams finish first, second, and third?

There are six baseball teams in the American League's Eastern division. In how many ways can:} -all six teams finish first through sixth?

A new campus project requires a committee of 5 students which will include 2 seniors and 3 juniors. There are 6 available seniors and 8 available juniors. -In how many ways can the juniors be chosen for the committee?

A new campus project requires a committee of 5 students which will include 2 seniors and 3 juniors. There are 6 available seniors and 8 available juniors. -In how many ways can the required committee be chosen?

The probability of selecting a defective component from 20 components, 3 of which are defective, is $\frac{3}{20}$ . This is an example of

Which of the following is not true of the symbol $\left(\begin{array}{c}11 \\ 3\end{array}\right)$ ?

In the past eight Friday nights, a student has gone to the same rock music club on five occasions. The claim that the probability is $\frac{5}{8}$ that the student will attend the same club next Friday is based on