Problem 77 - Algebra 2

Question

Find the value of \((3+2) !-(4-2) !\)

Step-by-Step Solution

Answer

The value of \((3+2)!-(4-2)!\) is 118.

Step 1: Understanding Factorial

Recognize that the '!' symbol represents a factorial, meaning the product of all positive integers less than or equal to the number. For example, 5! = 5 × 4 × 3 × 2 × 1.

Step 2: Simplify the Expressions Within Parentheses

Evaluate the expressions within the parentheses before addressing the factorial. Thus, calculate (3+2) and (4-2) to get 5 and 2, respectively.

Step 3: Calculate the Factorials

Compute the factorials of the two numbers obtained from the previous step: 5! = 5 × 4 × 3 × 2 × 1 = 120 and 2! = 2 × 1 = 2.

Step 4: Perform the Subtraction

Subtract the second factorial from the first: 120 - 2 = 118.

Key Concepts

Factorial Notation

Factorial notation is a mathematical concept used to describe the product of an integer and all the non-zero integers below it. For example, the factorial of 4, denoted as 4!, is calculated by multiplying 4 by 3, then by 2, and finally by 1. This leads to the equation:
\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \].
In general, if \(n\) is a positive integer, the factorial of \(n\) is expressed as \(n!\) and is the product of all positive integers from \(n\) down to 1. Factorial notation comes into play in various areas of mathematics, including permutations and combinations, which are fundamental in probability and statistics.

Simplifying Mathematical Expressions

To simplify a mathematical expression means to reduce it to its simplest form, making it easier to understand or solve. The process often involves several steps, such as performing arithmetic operations inside parentheses first, combining like terms, and applying mathematical properties like the distributive or associative properties.
When faced with an expression involving factorials, like in the exercise \( (3+2)! - (4-2)! \), the first step is to simplify the parts of the expression inside the parentheses. This makes the equation more manageable and sets the stage for applying factorial notation and further arithmetic operations. In our example, simplifying the expressions inside the parentheses leads to 5! and 2!, which are more straightforward when calculating their factorial values.

Arithmetic Operations

Arithmetic operations are the foundation of basic mathematics. They include addition, subtraction, multiplication, division, and the calculation of factorials. Each operation has a specific order of operations, commonly known as PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction).
In the context of the given exercise, after simplifying the parentheses and calculating the factorials, the final step is the subtraction of one factorial from another. This arithmetic operation gives us the final answer to the problem. Mastering the order and method of these operations is crucial when simplifying mathematical expressions involving factorials, as it ensures accurate computation and establishes a solid foundation for more advanced mathematical problem-solving.