Problem 1 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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

Step-by-Step Solution

Answer

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 1: 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: \[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

Step 2: 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: \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \]

Step 3: 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: \[ 3 \cdot 120 = 360 \]

Step 4: 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.

Key Concepts

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.