1. **State the problem:** Calculate the value of the expression $10! - \binom{6}{1} \times 2$. 2. **Recall the formulas:** - Factorial: $n! = n \times (n-1) \times \cdots \times 1$ - Binomial coefficient: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ 3. **Calculate each part:** - Calculate $10!$: $$10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3628800$$ - Calculate $\binom{6}{1}$: $$\binom{6}{1} = \frac{6!}{1! \times 5!} = \frac{6 \times 5!}{1 \times 5!} = 6$$ 4. **Multiply $\binom{6}{1}$ by 2:** $$6 \times 2 = 12$$ 5. **Subtract the product from $10!$:** $$3628800 - 12 = 3628788$$ **Final answer:** $$3628788$$