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In the realm of mathematics, functions serve as the fundamental building blocks which pair inputs to outputs in a consistent and rule-based manner. This characteristic means that for any given input, there is a definite output.

When dealing with factorial functions, the rule is that an input, say 'n', is mapped to the product of all positive integers less than or equal to 'n'. In formal terms, this is denoted as 'n!'. So, when you encounter '5!', this corresponds to the product 5 x 4 x 3 x 2 x 1. Factorials grow very quickly in size with the increase of the input number and are frequently employed in fields such as combinatorics, algebra, and calculus to solve problems involving permutations, combinations, and series.

Factorial division refers to the process of dividing one factorial by another. While at first glance this might seem like a daunting task given the large size of factorials, in practice it is often greatly simplified by cancelling out common terms.

For instance, when we see an expression like \(\frac{600!}{599!}\), it may look complex; however, understanding that every term in 599! is also a part of 600! allows for simplification. By recognizing that 599! contains all the factors of 600! except for 600 itself, we can cancel out these common factors, leaving us with just the extra term from 600!, which is 600. Thus, factorial division often involves identifying and eliminating these common factors to simplify the expression.

The process of simplifying factorials is a critical skill in tackling problems efficiently. When simplifying expressions like \(\frac{600!}{599!}\), where one factorial is a subset of the other, cancellation is your main tool. This is because factorials are sequential products; they contain within them the product of all preceding integers down to 1.

In high-level terms, when simplifying such an expression, focus on the fact that you're essentially removing the overlap between the numerator and the denominator. Remember, simplification is all about making expressions more manageable and easier to work with. When confronting factorial expressions, always look for opportunities to cancel out common factors, leading to a much simpler form, as you saw with \(\frac{600!}{599!}\) equals 600.