Answer: 24 zeroes.

Explanation:
When yo want to know how many times the digit 0 appears at the end of the number X, you have to remember that the answer is exactly the number of times that X is divisible by 10.
The meaning of ” X is divisible by 10″ is that X is divisible by 2 and 5 together. In the list of numbers from 1 to 100, every even number is divisible by 2 and only every fifth number is divisible by 5. Therefore, the number
of times X is divisible by 5 is less than the number of times X is divisible by 2. Therefore the number of times X is divisible by 5 is what determines the number of times it is divisible by 10. Hence, if we find how many times 5 participates in the product that creates 100!, we will get the solution.

Let’s look at the multiplication of the numbers from 1 to 100 and start replacing the number 5 with 5*1, the number 10 with 5*2  and so on.  Thus we will get 20 occurrences of 5.  But we must notice that in the four numbers 25, 50, 75, 100 the number 5 participates not once but twice (25 = 5*5, 50 = 5*5*2, 75 = 5*5*3, 100 =  5*5*4) so we have 4 more fives so in total we have 5 participating 24 times.