Answer: Two zeros


You could write this problem out as $1\times3\times4\times6\times7\times8\times9\times(2\times5)\times10$. The product of the first seven numbers is not a multiple of ten, but the last part gives us $10\times10$ so the overall product must be a multiple of $100$, but not a multiple of $1000$. So the answer is two zeros.