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Chapter 3: Problem 14

Five of ten books are arranged on a shelf. In how many ways can this be done?

Short Answer

Expert verified
The five books can be arranged in 30,240 ways.

Step by step solution

01

Identify n and r

In this problem, there are 10 books in total and 5 books are to be arranged on the shelf. So, \( n = 10 \) and \( r = 5 \)
02

Apply the Permutation Formula

Substitute \( n = 10 \) and \( r = 5 \) into the permutation formula. \( P(10, 5) = \frac{10!}{(10-5)!} \)
03

Simplify the Factorials

Simplify \( 10! \) and \( (10-5)! \). Factorial of 10 is the product of all positive integers from 1 to 10. Factorial of 5 is the product of all positive integers from 1 to 5.
04

Calculate

Divide \( 10! \) by \( 5! \) to get the number of ways the five books can be arranged on the shelf

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Key Concepts

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

Factorial
When we talk about the permutation of items, the term 'factorial' often comes into play. A factorial, denoted by an exclamation mark (!), is a way to express the product of all positive integers from 1 to a given number. For example, when we say 5!, we mean 5 times 4 times 3 times 2 times 1, which equals 120. This is fundamental in calculating how many ways items can be arranged or ordered.

The concept of factorial grows rapidly as the number increases. For instance:
  • 3! = 3 脳 2 脳 1 = 6
  • 4! = 4 脳 3 脳 2 脳 1 = 24
  • 10! = 10 脳 9 脳 8 脳 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1 = 3,628,800

Factorials are a crucial part of permutations, where order matters in the arrangement of objects. Understanding how to compute a factorial is an essential foundation for solving permutation problems.
Combination
Combinations refer to the selection of items from a larger pool, where order does not matter. Unlike permutations, we are only interested in which items are selected, not the sequence they appear in.

For example, if you are choosing 3 books from a collection of 5, you would use combinations. The formula used for combinations is:\[ C(n, r) = \frac{n!}{r!(n-r)!} \]where:
  • \( n \) is the total number of items.
  • \( r \) is the number of items to choose.

This formula allows you to calculate how many distinct groups of items can be formed. Even though combinations are not directly used in arrangements where order matters, they help highlight the difference and importance of choosing versus arranging.
Mathematical Problem Solving
Mathematical problem solving is a broader skill that involves understanding the problem, identifying the knowns and unknowns, and applying the appropriate mathematical techniques to find a solution.

In our example of arranging books, the process involves several steps which follow a logical progression:
  • Identify the problem requirements, like how many items need arranging.
  • Choose the correct formula, in this case, permutation because order matters.
  • Apply formulas and simplify using mathematical operations such as factorials.

Breaking down problems into smaller, manageable steps helps in solving complex problems systematically. Developing strong problem-solving skills is essential not just for mathematics but also for tackling everyday challenges logically and efficiently.

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Most popular questions from this chapter

Compute how many 7 -digit numbers can be made from the digits 1,2,3,4,5,6,7 if there is no repetition and the odd digits must appear in an unbroken sequence.

Use the binomial theorem to show \(\sum_{k=0}^{n}\left(\begin{array}{l}n \\\ k\end{array}\right)=2^{n}\).

Is the following statement true or false? Explain. If \(A_{1} \cap A_{2} \cap A_{3}=\varnothing\), then \(\left|A_{1} \cup A_{2} \cup A_{3}\right|=\left|A_{1}\right|+\left|A_{2}\right|+\left|A_{3}\right|\)

How many 6 -digit numbers are even or are divisible by \(5 ?\)

The first five rows of Pascal's triangle appear in the digits of powers of \(11: 11^{0}=1\) \(11^{1}=11,11^{2}=121,11^{3}=1331\) and \(11^{4}=14641 .\) Why is this so? Why does the pattern not continue with \(11^{5} ?\)
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