The best way to evaluate $\frac{800!}{799!}$ without a calculator is to recognize that factorial expressions simplify by canceling out common terms. The factorial $800!$ can be expanded as follows:

$800! = 800 \times 799!$

This allows us to rewrite the fraction:

$\frac{800!}{799!} = \frac{800 \times 799!}{799!}$

Now, we can cancel $799!$ from the numerator and the denominator:

$\frac{800 \times 799!}{799!} = 800$

So, the simplified result is $800$.

Correct Answer:

A. Rewrite $800!$ as $800 \cdot (799!)$ and then cancel out $799!$ from the numerator and denominator.

Would you like further details on factorial simplification? Here are some related questions:

1. How does simplifying factorial expressions help in evaluating large factorial ratios?
2. What are some common mistakes when simplifying factorial expressions?
3. How can factorial simplification be applied to expressions with multiple factorial terms?
4. What is the difference between evaluating $\frac{n!}{(n-1)!}$ and $\frac{n!}{(n-2)!}$?
5. Can factorial expressions be simplified further when they involve powers of factorial terms?

Tip: Remember, for expressions like $\frac{n!}{(n-1)!}$, simplify by expanding only the necessary part of the factorial in the numerator to cancel terms directly.