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Factorials are fundamental in combinatorics, algebra, and mathematical analysis. The factorial of a non-negative integer 'n', denoted by 'n!', is the product of all positive integers less than or equal to 'n'. But how exactly does this work?

Consider '5!'. This is equal to 5 x 4 x 3 x 2 x 1, which equals 120. The properties of factorials become particularly helpful when dealing with division of factorials.

One key property is that for any positive integer 'n', 'n!' can be represented as 'n' times '(n-1)!'. Mathematically, this is expressed as:
\[ n! = n \times (n - 1)! \]
This property simplifies the division of two factorials where one is the factorial of the other's consecutive number, such as \( \frac{n!}{(n-1)!} \). Here, all terms in '(n-1)!' cancel out the same terms in 'n!', leaving us with 'n' as the result. Understanding and utilizing this property allows for effortless simplification of factorial expressions, even without a calculator.

When tackling mathematical problems, a systematic approach can be the difference between a clear solution and confusion. Effective problem-solving involves understanding the problem, breaking it down into manageable parts, and applying mathematical concepts to arrive at a solution.

In the case of factorial simplification, as seen in the example \( \frac{900!}{899!} \), the problem-solving process involves recognizing that a factorial represents a series of multiplications. Conceptualizing '900!' as 900 x 899 x 898 x ... x 2 x 1 allows us to see that '899!' is embedded in '900!'—it's 899 x 898 x ... x 2 x 1.

This insight guides us to apply factorial properties effectively. Always remember that mathematical problem-solving is not just about getting the answer but understanding the 'why' and 'how' behind each step – this builds a stronger foundation for dealing with more complex problems.

Calculators are a go-to tool for many mathematical computations, but there are instances where evaluating expressions without one not only is possible but also better aids comprehension. When working without a calculator, it's essential to become comfortable with basic arithmetic skills and number sense.

For instance, grasping how to handle expressions involving factorials means recognizing patterns and employing shortcuts based on factorial properties. When evaluating \( \frac{900!}{899!} \) without a calculator, instead of computing the vast numbers, we apply the understanding of factorial division. Realizing that '899!' cancels the majority of the product in '900!', we are left with the last multiplication, which is simply 900. This demonstrates a beautiful aspect of mathematics: complex expressions can often be simplified using logic and properties.

Building this number sense and familiarity with mathematical properties enhances your ability to tackle expressions that might seem daunting at first glance, equipping you to solve a wide range of problems with confidence.