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The factorial of a non-negative integer, represented by an exclamation point (!), is a foundational concept in mathematics. It's the product of all positive integers up to a given number. For instance, the factorial of 5, written as 5!, is calculated by multiplying all positive integers from 1 up to 5, so it's equal to 1 × 2 × 3 × 4 × 5, which results in 120.

When dealing with larger numbers, manual calculation of factorials becomes cumbersome, and this is where the factorial key on a calculator becomes invaluable. By simply inputting the number and pressing the factorial key, you instantly obtain the result. It's important to note that the factorial function is only defined for non-negative integers. For instance, in the exercise \(\left(\frac{{300}}{{20}}\right)!\), the division is conducted first, reducing it to 15!, and then the factorial key is used to find the product of all positive integers from 1 to 15.

Modern calculators are equipped with a variety of functions that make complex mathematical operations simpler. In addition to basic arithmetic operations, calculators often have keys specifically for square roots, exponents, and factorials, among others. Understanding how to use these functions can save you a significant amount of time and reduce potential errors.

When tackling the given exercise, you would first use the calculator's division function to find the quotient of 300 divided by 20. Afterward, the factorial key can be used to compute the factorial of the result. Calculators with a factorial function allow direct computation, eliminating the need to multiply each consecutive integer manually.

Arithmetic operations such as addition, subtraction, multiplication, and division are the building blocks of mathematics. They are used in a sequence to solve complex problems, and the order in which they are performed can drastically affect the result. This sequence is guided by the rules of arithmetic, commonly known as 'order of operations' or PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

In the context of our exercise, the division is the first operation because it's inside the parentheses. After obtaining the result, we proceed to the next operation, which, in this case, is the factorial. Adhering to the correct order of operations ensures accuracy in mathematical problems, and leveraging the functions of a calculator can greatly expedite the process, allowing us to concentrate on understanding the problem rather than laboring over calculations.