Fast Factorial

SInce the first factorial with 3 trailing zeros is $15!=1307674368000$ , any factorials after $14!$ will not change the last 3 digits of $\sum_{n=1}^{100}{n!}$ .

Knowing this, it becomes clear that taking $\sum_{n=1}^{14}{n!} \pmod{1000}$ (trivial arithmetic) gives the sought after answer of $\boxed{313}$ .