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Chapter 5: Problem 36

A certain digital music player randomly plays each of 10 songs. Once a song is played, it is not repeated until all the songs have been played. In how many different ways can the player play the 10 songs?

Short Answer

Expert verified
There are 3,628,800 different ways the player can play the 10 songs.

Step by step solution

01

Understanding the problem

First, understand that the problem is asking for the number of possible orders in which 10 songs can be played without repetition.
02

Define the concept of permutation

Recognize that the problem is related to permutation because it involves arranging a set number of unique items (in this case, songs) in a sequence. A permutation of n distinct items is an arrangement of those items in a particular order.
03

Formula for permutation

The general formula for the number of permutations of n distinct items is given by n! where '!' denotes factorial, which means multiplying a series of descending natural numbers.
04

Calculate the factorial

For this specific problem, n = 10. Hence, calculate 10! which is: 10! = 10 脳 9 脳 8 脳 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1
05

Perform the multiplication

Multiply the numbers together to find the factorial: 10! = 3,628,800
06

Write down the final answer

The number of different ways the digital music player can play the 10 songs is 3,628,800.

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Key Concepts

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

factorial
A factorial, represented by an exclamation mark (!), is the product of all positive integers up to a certain number. For example, the factorial of 5, written as 5!, is calculated as 5 脳 4 脳 3 脳 2 脳 1.

Factorials are fundamental in the study of permutations and combinations, as they help count the ways to arrange or select items. For instance, calculating 10! involves multiplying all whole numbers from 10 down to 1.
For our specific problem with 10 songs, 10! = 10 脳 9 脳 8 脳 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1 = 3,628,800.
arranging distinct items
Arranging distinct items means ordering items in different sequences when each item is unique. In our example with 10 songs, every song is different, so we need to determine how many different orders or arrangements are possible.

To find this, we use permutations since each arrangement is a unique sequence. The formula to calculate the number of permutations of n distinct items is n!.
Since each song must be played once before any repeats, the digital player can arrange 10 unique songs in 10! ways.
This ensures every possible sequence is counted, leading to a total of 3,628,800 distinct arrangements.
combinatorics
Combinatorics is a branch of mathematics focused on counting, arranging, and combining items. It plays a crucial role in solving problems involving permutations and combinations.

In our exercise, we are specifically dealing with permutations. Permutations address the number of ways to order items, where the sequence matters. The formula used here, n!, directly comes from combinatorial principles.
By understanding combinatorics, you can solve complex problems involving various arrangements and selections. For our music player example, understanding permutations allowed us to calculate the 3,628,800 different ways to play 10 songs.

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Most popular questions from this chapter

Suppose that there are 55 Democrats and 45 Republicans in the U.S. Senate. A committee of seven senators is to be formed by selecting members of the Senate randomly. (a) What is the probability that the committee is composed of all Democrats? (b) What is the probability that the committee is composed of all Republicans? (c) What is the probability that the committee is composed of three Democrats and four Republicans?

You are dealt 5 cards from a standard 52-card deck. Determine the probability of being dealt three of a kind (such as three aces or three kings) by answering the following questions: (a) How many ways can 5 cards be selected from a 52-card deck? (b) Each deck contains 4 twos, 4 threes, and so on. How many ways can three of the same card be selected from the deck? (c) The remaining 2 cards must be different from the 3 chosen and different from each other. For example, if we drew three kings, the 4th card cannot be a king. After selecting the three of a kind, there are 12 different ranks of card remaining in the deck that can be chosen. If we have three kings, then we can choose twos, threes, and so on. Of the 12 ranks remaining, we choose 2 of them and then select one of the 4 cards in each of the two chosen ranks. How many ways can we select the remaining 2 cards? (d) Use the General Multiplication Rule to compute the probability of obtaining three of a kind. That is, what is the probability of selecting three of a kind and two cards that are not like?

In airline applications, failure of a component can result in catastrophe. As a result, many airline components utilize something called triple modular redundancy. This means that a critical component has two backup components that may be utilized should the initial component fail. Suppose a certain critical airline component has a probability of failure of 0.006 and the system that utilizes the component is part of a triple modular redundancy. (a) Assuming each component's failure/success is independent of the others, what is the probability all three components fail, resulting in disaster for the flight? (b) What is the probability at least one of the components does not fail?

According to the U.S. Census Bureau, \(20.2 \%\) of American women aged 25 years or older have a Bachelor's Degree; \(16.5 \%\) of American women aged 25 years or older have never married; among American women aged 25 years or older who have never married, \(22.8 \%\) have a Bachelor's Degree; and among American women aged 25 years or older who have a Bachelor's Degree, \(18.6 \%\) have never married. (a) Are the events "have a Bachelor's Degree" and "never married" independent? Explain. (b) Suppose an American woman aged 25 years or older is randomly selected, what is the probability she has a Bachelor's Degree and has never married? Interpret this probability.

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