jmc

The prime factorization of $200,000$ is $2^6\cdot 5^5$. Then count the number of factors of $2$ and $5$ in $20!$. Since there are $10$ even numbers, there are more than $6$ factors of $2$. There are $4$ factors of $5$. So the greatest common factor is $2^6\cdot 5^4=\boxed{40,000}$.