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Chapter 7: Problem 1

Evaluate the given expression. $$ 3 \cdot 5 ! $$

Short Answer

Expert verified
The given expression \(3 \cdot 5 !\) evaluates to 360 after calculating the factorial part (\(5! = 120\)) and performing the multiplication (\(3 \cdot 120 = 360\)).

Step by step solution

01

Understand the Factorial Function

The factorial function is a mathematical operation that takes a non-negative integer as input and returns the product of all positive integers less than or equal to that integer. The factorial function is denoted by the symbol "!". For example, if we have the expression 4!, it reads "Four Factorial" and is calculated as follow: \[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]
02

Calculate the Factorial Part of the Expression

In the given expression \(3 \cdot 5 !\), we first need to calculate the factorial part, which is \(5!\). Using the factorial function definition: \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]
03

Perform the Multiplication

Now that we have calculated the factorial part of the expression, we can perform the multiplication: \[ 3 \cdot 5! = 3 \cdot 120 \] Multiply the two numbers: \[ 3 \cdot 120 = 360 \]
04

Write the Final Answer

After completing the multiplication, the evaluated expression is: \[ 3 \cdot 5! = 360 \] So, the given expression \(3 \cdot 5 !\) evaluates to 360.

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Key Concepts

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

Mathematical Operations
In mathematics, operations are fundamental processes that allow us to solve mathematical expressions and equations. The most common mathematical operations include addition, subtraction, multiplication, and division. Each operation follows specific rules and properties that dictate how numbers interact with each other.
  • Addition combines two numbers to get a sum.
  • Subtraction finds the difference between two numbers.
  • Multiplication is the repeated addition of a number as many times as the second number indicates.
  • Division splits a number into equal parts as dictated by another number.
In the given exercise, understanding how to perform multiplication is crucial as it is a core part of solving the factorial and the final expression.
Integer Factorial
The factorial function is a fascinating mathematical operation that serves various practical purposes. It's primarily used in permutations, combinations, and in the analysis of algorithms. The factorial of a non-negative integer, represented by the symbol "!", is the result of multiplying that number by every positive integer less than itself. For example,
if you look at \(4!\) (read as "four factorial"), it means:
  • Start at 4
  • Multiply by 3, then 2, then 1
  • This gives \(4! = 4 \times 3 \times 2 \times 1 = 24\)
The concept applies to any integer\(n\), so \(n! = n \times (n-1) \times \ldots \times 1\). This makes the factorial operation a key part of resolving the initial step in the given problem, where you had to discover the result of \(5!\) as part of the solution.
Multiplication in Mathematics
Multiplication in mathematics is a powerful operation that allows us to efficiently add groups of numbers. When you multiply, you essentially add a number to itself repeatedly. For example, multiplying 3 by 4 is the same as adding three fours together: \(3 \times 4 = 4 + 4 + 4 = 12\).

In the context of the factorial evaluation, multiplication occurs multiple times:
  • First, during the factorial calculation, as shown with \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\), each number is multiplied by the one before it.
  • Second, in the final step of the problem, where the result of the factorial (\(120\)) is multiplied by 3 to obtain \(3 \times 120 = 360\).
Understanding multiplication's role helps streamline the process of breaking down mathematical expressions step-by-step, making it easier to solve problems efficiently.

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Most popular questions from this chapter

A time study was conducted by the production manager of Vista Vision to determine the length of time in minutes required by an assembly worker to complete a certain task during the assembly of its Pulsar color television sets. a. Describe a sample space corresponding to this time study. b. Describe the event \(E\) that an assembly worker took 2 min or less to complete the task. c. Describe the event \(F\) that an assembly worker took more than 2 min to complete the task.

An experiment consists of selecting a card at random from a well-shuffled 52 -card deck. Let \(E\) denote the event that an ace is drawn and let \(F\) denote the event that a spade is drawn. Show that \(n(E \cup F)=n(E)+n(F)-n(E \cap F)\).

In a television game show, the winner is asked to select three prizes from five different prizes, \(A, B\), \(\mathrm{C}, \mathrm{D}\), and \(\mathrm{E} .\) a. Describe a sample space of possible outcomes (order is not important). b. How many points are there in the sample space corresponding to a selection that includes A? c. How many points are there in the sample space corresponding to a selection that includes \(\mathrm{A}\) and \(\mathrm{B}\) ? d. How many points are there in the sample space corresponding to a selection that includes either \(\mathrm{A}\) or \(\mathrm{B}\) ?

List the simple events associated with each experiment. Data concerning durable goods orders are obtained each month by an economist. A record is kept for a 1 -yr period of any increase \((i)\), decrease \((d)\), or unchanged movement \((u)\) in the number of durable goods orders for each month as compared with the number of such orders in the same month of the previous year.

According to a study of 100 drivers in metropolitan Washington, D.C., whose cars were equipped with cameras with sensors, the distractions and the number of incidents (crashes, near crashes, and situations that require an evasive maneuver after the driver was distracted) caused by these distractions are as follows: $$ \begin{array}{lccccccccc} \hline \text { Distraction } & A & B & C & D & E & F & G & H & I \\ \hline \text { Driving Incidents } & 668 & 378 & 194 & 163 & 133 & 134 & 111 & 111 & 89 \\ \hline \end{array} $$ where \(A=\) Wireless device (cell phone, PDA) \(\begin{aligned} B &=\text { Passenger } \\ C &=\text { Something inside car } \\ D &=\text { Vehicle } \\ E &=\text { Personal hygiene } \\ F &=\text { Eating } \\ G &=\text { Something outside car } \\ H &=\text { Talking/singing } \\ I &=\text { Other } \end{aligned}\) If an incident caused by a distraction is picked at random, what is the probability that it was caused by a. The use of a wireless device? b. Something other than personal hygiene or eating?
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