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Chapter 4: Problem 19

Permutations Evaluate the following permutations. a. \(P_{3}^{5}\) b. \(P_{9}^{10}\) c. \(P_{6}^{6}\) d. \(P_{1}^{20}\)

Short Answer

Expert verified
Question: What are the values of the following permutations: a. \(P_{3}^{5}\) b. \(P_{9}^{10}\) c. \(P_{6}^{6}\) d. \(P_{1}^{20}\) Answer: a. \(P_{3}^{5} = 60\) b. \(P_{9}^{10} = 0\) c. \(P_{6}^{6} = 720\) d. \(P_{1}^{20} = 20\)

Step by step solution

01

a. Evaluate \(P_{3}^{5}\)

First, identify the values of n and r. In this case, n = 5, and r = 3. Now, use the permutation formula \(P_{n}^{r} = \frac{n!}{(n-r)!}\) Plug in the values of n and r: \(P_{3}^{5} = \frac{5!}{(5-3)!}\) Now calculate the factorials: \(P_{3}^{5} = \frac{120}{2!} = \frac{120}{2} = 60\) The value of \(P_{3}^{5}\) is 60.
02

b. Evaluate \(P_{9}^{10}\)

Since r is greater than n in this case (r = 10 > n = 9), this permutation is not possible, as there aren't enough elements to arrange. Therefore, \(P_{9}^{10} = 0\).
03

c. Evaluate \(P_{6}^{6}\)

First, identify the values of n and r. In this case, n = 6, and r = 6. Now, use the permutation formula \(P_{n}^{r} = \frac{n!}{(n-r)!}\) Plug in the values of n and r: \(P_{6}^{6} = \frac{6!}{(6-6)!}\) Now calculate the factorials: \(P_{6}^{6} = \frac{720}{0!} = \frac{720}{1} = 720\) The value of \(P_{6}^{6}\) is 720.
04

d. Evaluate \(P_{1}^{20}\)

First, identify the values of n and r. In this case, n = 20, and r = 1. Now, use the permutation formula \(P_{n}^{r} = \frac{n!}{(n-r)!}\) Plug in the values of n and r: \(P_{1}^{20} = \frac{20!}{(20-1)!}\) Now calculate the factorials: \(P_{1}^{20} = \frac{2432902008176640000}{121645100408832000} = 20\) The value of \(P_{1}^{20}\) is 20.

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Key Concepts

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

Factorial
The factorial of a non-negative integer, denoted by an exclamation mark (!), is the product of all positive integers less than or equal to that number. For example, the factorial of 5, written as 5!, is calculated as 5 脳 4 脳 3 脳 2 脳 1, which equals 120. Importantly, by definition, the factorial of 0 (0!) is always 1.

This concept is foundational in permutations as it represents the number of ways to arrange a set of objects. When calculating factorials, remember that the value increases exponentially with the size of the number, so factorials can become very large very quickly. For computational purposes in permutations, we often use the factorial to determine the number of different arrangements or sequences that can be formed from a set of distinct objects.
Permutation Formula
Permutations refer to the number of ways to arrange a subset of items from a larger set, where the order is important. The permutation formula, denoted as \( P_n^r \), gives us a way to calculate this. Here, \(n\) represents the total number of items to choose from, and \(r\) is the number of items to arrange. The formula is:\[ P_n^r = \frac{n!}{(n-r)!} \]

In essence, this formula takes the total number of arrangements of \(n\) items and divides it by the number of arrangements of the items that are not included in each arrangement (\(n - r\) items). It's worth noting that if \(r > n\), the permutation value is zero because it's impossible to arrange more items than what are available. Understanding this formula is key to solving problems involving the order of arrangements, which are common in combinatorics.
Combinatorics
Combinatorics is an area of mathematics concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is used in various fields, including computer science, statistics, and physics. Combinatorics is related to other areas of mathematics, such as algebra, probability, and geometry, through various topics like graph theory, configurations, and packing and covering problems.

The study of permutations is a fundamental aspect of combinatorics, as it involves the arrangement of items in a particular order. Mastering permutations and related concepts, like the factorial and the permutation formula, provides a strong foundation for further exploration in combinatorial problems. These concepts help in solving puzzles, optimizing solutions, and analyzing possible outcomes, highlighting the importance of combinatorics in problem-solving and logical thinking.

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Most popular questions from this chapter

During the inaugural season of Major League Soccer in the United States, the medical teams documented 256 injuries that caused a loss of participation time to the player. The results of this investigation, reported in The American Journal of Sports Medicine, is shown in the table. \({ }^{3}\) If one individual is drawn at random from this group of 256 soccer players, find the following probabilities: a. \(P(A)\) b. \(P(G)\) c. \(P(A \cap G)\) d. \(P(G \mid A)\) e. \(P(G \mid B)\) f. \(P(G \mid C)\) g. \(P(C \mid P)\) h. \(P\left(B^{c}\right)\)

A company has five applicants for two positions: two women and three men. Suppose that the five applicants are equally qualified and that no preference is given for choosing either gender. Let \(x\) equal the number of women chosen to fill the two positions. a. Find \(p(x)\). b. Construct a probability histogram for \(x\)

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A taste-testing experiment is conducted at a local supermarket, where passing shoppers are asked to taste two soft-drink samples-one Pepsi and one Coke-and state their preference. Suppose that four shoppers are chosen at random and asked to participate in the experiment, and that there is actually no difference in the taste of the two brands. a. What is the probability that all four shoppers choose Pepsi? b. What is the probability that exactly one of the four shoppers chooses Pepsi?

Three people are randomly selected from voter registration and driving records to report for jury duty. The gender of each person is noted by the county clerk. a. Define the experiment. b. List the simple events in \(S\). c. If each person is just as likely to be a man as a woman, what probability do you assign to each simple event? d. What is the probability that only one of the three is a man? e. What is the probability that all three are women?
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