Ch. 3 - Probability

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 3 - Probability

Problem 3.T.7b

Chapter 3, Problem 3.T.7b

7. There are 16 students giving final presentations in your history course.
b. Presentation subjects are based on the units of the course. Unit B is covered by three students, Unit C is covered by five students, and Units A and D are each covered by four students. How many presentation orders are possible when presentations on
the same unit are indistinguishable from each other?

Verified step by step guidance

Step 1: Understand the problem. The total number of students is 16, and they are grouped by units: Unit A (4 students), Unit B (3 students), Unit C (5 students), and Unit D (4 students). Presentations on the same unit are indistinguishable, meaning the order within each unit does not matter.

Step 2: Recall the formula for permutations of indistinguishable items. When arranging a set of items where some are indistinguishable, the total number of arrangements is given by the formula: \( \frac{n!}{k_1! \cdot k_2! \cdot \dots \cdot k_m!} \), where \( n \) is the total number of items, and \( k_1, k_2, \dots, k_m \) are the counts of indistinguishable items in each group.

Step 3: Apply the formula. Here, \( n = 16 \) (total students), and the groups are \( k_1 = 4 \) (Unit A), \( k_2 = 3 \) (Unit B), \( k_3 = 5 \) (Unit C), and \( k_4 = 4 \) (Unit D). Substitute these values into the formula: \( \frac{16!}{4! \cdot 3! \cdot 5! \cdot 4!} \).

Step 4: Simplify the factorials. Compute the factorials for \( 16! \), \( 4! \), \( 3! \), and \( 5! \), but do not calculate the final result yet. The factorial of a number \( n! \) is the product of all positive integers from 1 to \( n \). For example, \( 4! = 4 \cdot 3 \cdot 2 \cdot 1 \).

Step 5: Divide \( 16! \) by the product of \( 4! \cdot 3! \cdot 5! \cdot 4! \). This will give the total number of possible presentation orders where presentations on the same unit are indistinguishable. Leave the result in its simplified form if needed.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Permutations and Combinations

Permutations and combinations are fundamental concepts in combinatorics that deal with counting arrangements and selections. Permutations refer to the different ways to arrange a set of items where order matters, while combinations refer to selections where order does not matter. In this question, since presentations on the same unit are indistinguishable, combinations are more relevant for calculating the total arrangements.

Multinomial Coefficient

The multinomial coefficient is a generalization of the binomial coefficient and is used to count the ways to divide a set of items into multiple groups. It is expressed as n!/(k1! * k2! * ... * kr!), where n is the total number of items, and k1, k2, ..., kr are the sizes of the groups. In this scenario, it helps determine the number of distinct arrangements of students presenting on different units.

Indistinguishable Objects

Indistinguishable objects refer to items that cannot be differentiated from one another within a set. In this problem, the students presenting on the same unit are indistinguishable, meaning their order does not affect the overall arrangement. This concept is crucial for simplifying the calculation of possible presentation orders, as it reduces the total number of unique arrangements.

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Permutations of Non-Distinct Objects

Related Practice

Textbook Question

Your dorm enters 15 out of 65 plastic numbered ducks in a duck race. The ducks are all dumped into a stream and drift to the finish line. What is the probability that three of your dorm's ducks finish first, second, and third?

Textbook Question

5. Use technology to randomly select two numbers from 1 to 6. Find the sum and subtract 1 to obtain a total.

a. What is the theoretical probability of each total from 1 to 11?

b. Use this procedure to select 100 totals from 1 to 11. Tally your results and compare them with the probabilities in part (a).

Textbook Question

A person's building access code is their first and last initials and four digits.

You know a person's first name only, and you know that the last digit is odd. What is the probability of guessing this person's code on the first try?

Textbook Question

4. The table on the left shows the secondary school student enrollment levels (in thousands by grade) in Oklahoma and Texas schools in a recent year. (Source: U.S. Nation

for Education Statistics)

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Textbook Question

You work in the security department of a bank’s website. To access their accounts, customers of the bank must create an 8-digit password. It is your job to determine the password requirements for these accounts. Security guidelines state that for the website to be secure, the probability that an 8-digit password is guessed on one try must be less than 1/60^8, assuming all passwords are equally likely.

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3. For additional security, each customer creates a 5-digit PIN (personal identification number). The table on the right shows the 10 most commonly chosen 5-digit PINs. From the table, you can see that more than a third of all 5-digit PINs could be guessed by trying these 10 numbers. To discourage customers from using predictable PINs, you consider prohibiting PINs that use the same digit more than once.

b. Would you decide to prohibit PINs that use the same digit more than once? Explain.

Textbook Question

4. The table on the left shows the secondary school student enrollment levels (in thousands by grade) in Oklahoma and Texas schools in a recent year. (Source: U.S. Nation

for Education Statistics)

A student in one of the indicated grades and states is randomly selected. Find the probability of selecting a student who

a. is in ninth grade.