249
Edit: reworded for clarity
The number of zeroes following a factorial is equal to the number of 5 factors(a factorial has more 2 factors than five, so for every 5, there's a 2 to multiply it by, giving you a factor of 10, and therefore a zero on the end). So 10! has 2 trailing zeroes, from 5 and 10.
So simply dividing by 5 appears to work, but what happens when you get to 25? 25 is 5x5, so it will cause 2 trailing zeroes, as will every multiple of 25.
Then for every multiple of 125, you get 3 zeroes; and 625, 4.
so, 1001 is followed by (1001/5)+(1001/25)+(1001/125)+(1001/625)=249.