📘 algebra, combinatorics, number theory, geometry

1. **State the problem:** Evaluate the sum $$\sum_{k=0}^{2025} \frac{2025! \cdot (-1)^k 2^k}{k! (2025-k)!}$$ and determine which of the options A) $1^{-(2\sqrt{2025})}$ B) $-1$ C)