To solve the expression:

\[
\frac{10! \times 4!}{8! \times 6!}
\]

We start by simplifying the factorials:

- \(10! = 10 \times 9 \times 8!\)
- \(8! = 8!\)
- \(6! = 6!\)
- \(4! = 4 \times 3 \times 2 \times 1 = 24\)

Now, cancel out the common factorials:

\[
\frac{10 \times 9 \times 8! \times 4!}{8! \times 6!} = \frac{10 \times 9 \times 24}{6!}
\]

Next, simplify \(6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720\).

Now, substitute \(6! = 720\) and calculate the result:

\[
\frac{10 \times 9 \times 24}{720} = \frac{2160}{720} = 3
\]

Thus, the value of the expression is:

\[
\boxed{3}
\]

Would you like further details or have any questions?

Here are five related questions:
1. What is the factorial of a number?
2. How do you simplify expressions involving factorials?
3. Can factorials be applied to decimal or negative numbers?
4. What are the properties of permutations involving factorials?
5. How do you calculate large factorials efficiently?

**Tip:** Simplifying common terms in factorials before multiplication helps reduce the size of numbers for easier calculations!

Explore the step-by-step process of simplifying the factorial expression 10! 4! / 8! 6!. This problem involves the concepts of factorials and simplification using basic arithmetic. Perfect for students in Grades 9-12 looking to master factorial operations.

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