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Mathematical operations form the foundation of many calculations in mathematics. In this particular exercise, we are dealing with division. Division is one of the four basic arithmetic operations, alongside addition, subtraction, and multiplication.

Division involves distributing a total amount evenly into groups, and it is denoted by the division operator "÷" or a fraction bar "/". In our expression \( \left( \frac{12}{4} \right) \), we divide 12 by 4. Think of it as asking how many times the number 4 fits into 12. Since 4 goes into 12 exactly three times, the result of \( \frac{12}{4} \) is 3. Performing these operations correctly sets the stage for further steps in solving more complex expressions.

The order of operations is a set of rules that dictates the order in which mathematical expressions should be evaluated. This is crucial to obtaining accurate results, as different orders can lead to different outcomes.

The standard rule is often remembered by the acronym PEMDAS, which stands for:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In our exercise, we start with the parentheses first. We compute \( \left( \frac{12}{4} \right) \) before applying any other operations because operations inside parentheses have the highest precedence. Once we determine the value inside the parentheses, we can proceed to the next step, which involves applying the factorial operation.

Factorial calculation is an operation often encountered in mathematics, particularly in combinatorics and algebra. The factorial of a number, denoted by an exclamation point "!", is the product of all positive integers from 1 up to that number.

In our example, after computing the division to get 3, we then calculate the factorial of 3, represented as \(3!\). To solve \(3!\), we multiply 3 by every positive integer below it:

• Start with 3
• Then multiply by 2
• Finally multiply by 1

This results in the calculation: \(3 \times 2 \times 1 = 6\). Therefore, the factorial of 3 is 6. Understanding how to compute factorials is essential for solving mathematical problems involving combinations, permutations, and sequences.