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Chapter 5: Problem 13

Find the value of each permutation. $$ { }_{4} P_{4} $$

Short Answer

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Step by step solution

01

Understand the Permutation Notation

The notation \({ }_{n} P_{r}\) represents the number of permutations of n items taken r at a time. The formula is given by \({ }_{n} P_{r} = \frac{n!}{(n-r)!}\).
02

Identify the Values of n and r

In the given problem, \({ }_{4} P_{4}\), n = 4 and r = 4.
03

Apply the Permutation Formula

Substitute the values of n and r into the formula: \({ }_{4} P_{4} = \frac{4!}{(4-4)!}\).
04

Evaluate the Factorials

Compute \(4! \) and \(0! \): \(4! = 4 \times 3 \times 2 \times 1 = 24\) and \(0! = 1\).
05

Calculate the Permutation Value

Use the factorial values in the formula: \({ }_{4} P_{4} = \frac{4!}{0!} = \frac{24}{1} = 24\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Permutation Formula
Permutations are a key part of combinatorial mathematics, representing the different ways to arrange or order a set of items. The permutation formula \( { }_{n} P_{r} = \frac{n!}{(n-r)!} \) helps calculate the number of distinct arrangements of n items taken r at a time. This formula is highly valuable in solving problems where the order of selection matters.Let's break the formula down:
  • n is the total number of items.
  • r is the number of items being chosen and arranged.
  • n! (n factorial) is the product of all positive integers up to n.
  • (n-r)! ((n-r) factorial) is the product of all positive integers up to (n-r).
By dividing n! by (n-r)!, we effectively remove the arrangements of items that are not needed, giving us the total number of permutations.
Factorials
Factorials are fundamental to understanding permutations. A factorial, denoted by an exclamation mark (e.g., 4!), is the product of all positive integers up to a given number. For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).Here are a few key properties of factorials:
  • 0! is defined as 1.
  • Factorials grow very quickly; 5! = 120, 6! = 720, and so on.
In the context of the permutation formula, factorials help us calculate the total and the restricted number of arrangements, making them an indispensable tool in combinatorial mathematics.
Combinatorial Mathematics
Combinatorial mathematics is a branch of mathematics dealing with combinations, permutations, and the counting of objects. It is essential in fields like computer science, statistics, and game theory.In combinatorial mathematics:
  • Permutations involve arranging objects in specific order (order matters).
  • Combinations involve selecting objects without regard to order (order doesn't matter).
Permutations (\( { }_{n} P_{r} \)) can be particularly useful in problems like:
  • Arranging books on a shelf.
  • Scheduling tasks or events.
  • Forming codes or passwords.
Understanding these fundamental concepts equips you with the tools to tackle more complex problems in various domains.

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Most popular questions from this chapter

According to Internal Revenue Service records, \(6.42 \%\) of all household tax returns are audited. According to the Humane Society, \(39 \%\) of all households own a dog. Assuming dog ownership and audits are independent events, what is the probability a randomly selected household is audited and owns a dog?

In the U.S. Senate, there are 21 members on the Committee on Banking, Housing, and Urban Affairs. Nine of these 21 members are selected to be on the Subcommittee on Economic Policy. How many different committee structures are possible for this subcommittee?

A box containing twelve 40-watt light bulbs and eighteen 60-watt light bulbs is stored in your basement. Unfortunately, the box is stored in the dark and you need two 60-watt bulbs. What is the probability of randomly selecting two 60-watt bulbs from the box?

Go to www.pearsonhighered.com/sullivanstats to obtain the data file SullivanStatsSurveyI using the file format of your choice for the version of the text you are using. The data represent the results of a survey conducted by the author. The variable "Text while Driving" represents the response to the question, "Have you ever texted while driving?" The variable "Tickets" represents the response to the question, "How many speeding tickets have you received in the past 12 months?" Treat the individuals in the survey as a random sample of all U.S. drivers. (a) Build a contingency table treating "Text while Driving" as the row variable and "Tickets" as the column variable. (b) Determine the marginal relative frequency distribution for both the row and column variable. (c) What is the probability a randomly selected U.S. driver texts while driving? (d) What is the probability a randomly selected U.S. driver received three speeding tickets in the past 12 months? (e) What is the probability a randomly selected U.S. driver texts while driving or received three speeding tickets in the past 12 months?

Suppose that a poll is being conducted in the village of Lemont. The pollster identifies her target population as all residents of Lemont 18 years old or older. This population has 6494 people. (a) Compute the probability that the first resident selected to participate in the poll is Roger Cummings and the second is Rick Whittingham. (b) The probability that any particular resident of Lemont is the first person picked is \(\frac{1}{6494} .\) Compute the probability that Roger is selected first and Rick is selected second, assuming independence. Compare your results to part (a). Conclude that, when small samples are taken from large populations without replacement, the assumption of independence does not significantly affect the probability.
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