1. The problem is to create a challenging factorial problem for practice. 2. Factorials are denoted by $n!$ and represent the product of all positive integers from 1 to $n$. 3. Important rules: - $0! = 1$ - $n! = n \times (n-1)!$ 4. Here is a hard factorial problem: Simplify and evaluate $$\frac{10!}{8! \times 2!}$$ 5. Step-by-step solution: - Write out the factorials: $$10! = 10 \times 9 \times 8!$$ - Substitute into the expression: $$\frac{10 \times 9 \times 8!}{8! \times 2!}$$ - Cancel $8!$ from numerator and denominator: $$\frac{10 \times 9}{2!}$$ - Calculate $2! = 2 \times 1 = 2$ - Simplify: $$\frac{90}{2} = 45$$ 6. Final answer: $$45$$