/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 33 Suppose Dan is going to upload 1... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 33

Suppose Dan is going to upload 12 songs to his digital music player. In how many ways can the 12 songs be played without repetition?

Short Answer

Expert verified
There are 479,001,600 ways to play the 12 songs without repetition.

Step by step solution

01

- Understand the Problem

We need to find the number of ways to arrange 12 songs so that they are all played once without repetition. This means we are looking for the number of possible permutations of 12 items.
02

- Apply the Permutation Formula

The number of permutations of n distinct items is given by the factorial of n, denoted as n!. Since we have 12 songs, we need to calculate 12!.
03

- Calculate the Factorial

To find 12!, multiply all whole numbers from 12 down to 1: 12! = 12 脳 11 脳 10 脳 9 脳 8 脳 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1.
04

- Compute the Result

Perform the multiplication to find the value of 12!: 12! = 479,001,600.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

factorial
Factorials are fundamental in understanding permutations and combinations. A factorial is denoted by an exclamation mark (!) and is the product of all positive integers up to a given number. For example, the factorial of 4 (written as 4!) is calculated by multiplying 4 脳 3 脳 2 脳 1 to get 24.
Factorials grow very quickly, which is why even relatively small numbers can result in very large factorial values. For example, 12! (the factorial of 12) equals 479,001,600.
Factorials are essential in permutations, as they provide the total number of ways to arrange a set of items without repetition.
counting principles
Counting principles are mathematical rules used to count the number of ways an event can occur. There are two main counting principles: the addition principle and the multiplication principle.
  • **Addition Principle**: If one event can occur in 'm' ways and a second event can occur in 'n' ways, and these events cannot happen at the same time, then the number of ways either event can occur is m + n.
  • **Multiplication Principle**: If one event can occur in 'm' ways and a second event can occur in 'n' ways after the first event, then the number of ways both events can occur is m 脳 n.
In our exercise, the multiplication principle is crucial because arranging songs involves choosing one song out of many and repeating this process without repetition until all songs are chosen. Mathematically, it means calculating the factorial of the total number of items to find the permutations.
combinatorics
Combinatorics is the branch of mathematics dealing with combinations and permutations of objects. It focuses on counting, arrangement, and combination of sets of elements.
A **Permutation** is an arrangement of objects in a specific order, and order is important. For instance, arranging 12 different songs in every possible order requires the use of permutations. Calculating permutations often involves using factorials, as seen in the solution of our exercise.
Combinatorics also includes **Combinations**, where the order does not matter. But for the problem at hand, permutations are key since the sequence of songs plays a crucial role.
Understanding combinatorics is vital for solving complex counting problems in various fields such as probability, computer science, and more.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In the game Text Twist, six letters are given and the player must form words of varying lengths using the letters provided. Suppose that the letters in a particular game are ENHSIC. (a) How many different arrangements are possible using all 6 letters? (b) How many different arrangements are possible using only 4 letters? (c) The solution to this game has three 6 -letter words. To advance to the next round, the player needs at least one of the six-letter words. If the player simply guesses, what is the probability that he or she will get one of the six-letter words on their first guess of six letters?

List all the combinations of four objects \(a, b, c,\) and \(d\) taken two at a time. What is \({ }_{4} C_{2} ?\)

Suppose that you just received a shipment of six televisions and two are defective. If two televisions are randomly selected, compute the probability that both televisions work. What is the probability that at least one does not work?

Lingo In the gameshow Lingo, the team that correctly guesses a mystery word gets a chance to pull two Lingo balls from a bin. Balls in the bin are labeled with numbers corresponding to the numbers remaining on their Lingo board. There are also three prize balls and three red "stopper" balls in the bin. If a stopper ball is drawn first, the team loses their second draw. To form a Lingo, the team needs five numbers in a vertical, horizontal, or diagonal row. Consider the sample Lingo board below for a team that has just guessed a mystery word. $$ \begin{array}{|l|l|l|l|l|} \hline \mathbf{L} & \mathbf{I} & \mathbf{N} & \mathbf{G} & \mathbf{O} \\ \hline 10 & & & 48 & 66 \\ \hline & & 34 & & 74 \\ \hline & & 22 & 58 & 68 \\ \hline 4 & 16 & & 40 & 70 \\ \hline & 26 & 52 & & 64 \\ \hline \end{array} $$ (a) What is the probability that the first ball selected is on the Lingo board? (b) What is the probability that the team draws a stopper ball on its first draw? (c) What is the probability that the team makes a Lingo on their first draw? (d) What is the probability that the team makes a Lingo on their second draw?

My wife has organized a monthly neighborhood party. Five people are involved in the group: Yolanda (my wife), Lorrie, Laura, Kim, and Anne Marie. They decide to randomly select the first and second home that will host the party. What is the probability that my wife hosts the first party and Lorrie hosts the second? Note: Once a home has hosted, it cannot host again until all other homes have hosted.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.