HMMT 2001 U60

Find an $n$ such that $n!-(n-1)!+(n-2)!-(n-3)!+\cdots \pm 1$ ! is prime. Be prepared to justify your answer for $\left\{\begin{array}{c}n, \\ {\left[\frac{n+225}{10}\right],} \\ n \leq 25\end{array}\right.$ points, where $[N]$ is the greatest integer less than $N$.