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Chapter 16: Problem 24

A set of reference books consists of eight volumes numbered 1 through 8 . (a) In how many ways can the eight books be arranged so that Volume 8 is in the correct position on the shelf (i.e., the last one from left to right)? (b) In how many ways can the eight books be arranged so that Volumes 1 and 2 are in their correct positions on the shelf?

Short Answer

Expert verified
The answer to the first question is 5,040 ways and the second question is 720 ways.

Step by step solution

01

Analyzing the first question

To solve the first question, we need to arrange 7 books (Volume 1 to Volume 7) and fix the position for Volume 8. Since the books are distinguishable and have different volumes, they can be arranged in \(7!\) ways.
02

Calculating for the first question

By using the formula for permutation of n different things, \(n!\), we find that 7 volumes can be arranged in \(7!\) = 5,040 ways with Volume 8 fixed in the last position.
03

Analyzing the second question

For the second question, we need to arrange 6 books (Volume 3 to Volume 8) and fix the positions of Volume 1 and Volume 2. Again, since the books are distinguishable, they can be arranged in \(6!\) ways.
04

Calculating for the second question

Using the formula for permutation of n different things, we find that the remaining 6 volumes can be arranged in \(6!\) = 720 ways with Volume 1 and Volume 2 fixed in their correct positions.

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Key Concepts

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

Factorials
Factorials are an essential concept in combinatorics and play a significant role in solving arrangement problems. The factorial of a non-negative integer, denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). To give you an example:
  • \( 1! = 1 \)
  • \( 2! = 2 \times 1 = 2 \)
  • \( 3! = 3 \times 2 \times 1 = 6 \)
  • \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
Factorials count the number of ways to arrange \( n \) distinct items in a sequence. In our problem, calculating \( 7! \) and \( 6! \) allowed us to find how many different ways we could arrange the books with specific volumes fixed in their positions. Understanding this simple concept simplifies many combinatorial problems.
Permutations
Permutations are arrangements of objects in a specific order. When the order of items matters, we use permutations. The formula for permutations of \( n \) different items taken all at once is simply \( n! \). In the textbook exercise, permutations are used to find the number of ways to organize volumes of books. For instance:
  • When Volume 8 is fixed in its position, only Volumes 1 to 7 are left to arrange, which leads to \( 7! \) permutations.
  • Similarly, by fixing Volumes 1 and 2, we arrange Volumes 3 to 8, equating to \( 6! \) permutations.
So remember, permutations tell us how many ways we can arrange a set of items uniquely considering the sequence.
Arrangements
In combinatorics, arrangements refer to the ways in which a set of things can be organized. Arrangements can be straightforward, or they can have certain constraints, like fixing some items in a position while rearranging the others. This is what we see in our problem with fixing certain volumes in place. When considering arrangements:
  • For the books with Volume 8 fixed, it transforms the problem into arranging the rest of the 7 books, resulting in \( 7! \) possible ways.
  • When Volumes 1 and 2 are fixed, the challenge changes to arranging Volumes 3 through 8, equaling \( 6! \) ways.
Understanding different arrangement possibilities helps to address questions like these confidently. Arranging inherently means finding the permutation of items, especially when specific elements are fixed.

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Most popular questions from this chapter

The Tasmania State University glee club has 15 members. A quartet of four members must be chosen to sing at the university president's reception. Assume that the quartet is chosen randomly by drawing the names out of a hat. Find the probability that (a) Alice (one of the members of the glee club) is chosen to be in the quartet. (b) Alice is not chosen to be in the quartet. (c) the four members chosen for the quartet are Alice, Bert, Cathy, and Dale.

Using set notation, write out the sample space for each of the following random experiments: (a) Three runners \((A, B,\) and \(C\) ) are running in a race (assume that there are no ties). The observation is the order in which the three runners cross the finish line. (b) Four runners \((A, B, C,\) and \(D)\) are running in a qualifying race (assume that there are no ties). The top two finishers qualify for the finals. The observation is the pair of runners that qualify for the finals.

An honest coin is tossed three times in a row. Find the probability of each of the following events. (Hint: Do Exercise 11 first.) (a) \(E_{1}\); "the coin comes up heads exactly twice." (b) \(E_{2}:\) "all three tosses come up the same." (c) \(E_{3}:\) "exactly half of the tosses come up heads." (d) \(E_{4}\) : "the first two tosses come up tails"

A card is drawn at random out of a well-shuffled deck of 52 cards. Find the probability of each of the following events. (Hint: Do Exercise 14 first.) (a) \(E_{1}:\) "draw a queen." (b) \(E_{2}:\) "draw a heart." (c) \(E_{3}:{ }^{*}\) draw a face card." \(\left(\mathrm{A}^{\text {" }}\right.\) face" card is a jack, queen, or king.)

The sample spaces are too big to write down in full. In these exercises, you should describe the sample space either by describing a generic outcome or by listing some outcomes and then using the ... notation. In the latter case, you should write down enough outcomes to make the description reasonably clear. A student randomly guesses the answers to a multiplechoice quiz consisting of 10 questions. The observation is the student's answer \((A, B, C, D,\) or \(E)\) for each question. Describe the sample space.
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