We are only looking for trailing zeroes.

As we multiply for our solution, anytime we multiply the current result by ten, we will add one zero to the result.

If we can find all the numbers in 100! that have 10 as a factor, we will know how many zeroes we have.

100 * 99 * 98 * 97 ... reduces to
(10 * 10) * (11 * 3 * 3) * (7 * 7 * 2) ... 

10, 20, 30, 40 etc all have 10 as a factor, so each of these numbers will add one zero.

Additionally, numbers such as 5, 15, 25, 35, etc do not have ten as a factor, but there will be plenty of 2's in the series to convert these into ten's. For example 5 times any of the 2's in our series will give us a 10. 15 * 2 = 30, 25 * 2 = 50, etc. For each five that appears another zero will be added.

10 squared (100) also has one additional zero, much in the same way that five squared (25, 50, and 75) will contribute one additional zero. For example 25*4 = 100, 50*2 = 100 and 75*4 = 300, so for each factor of five squared there is an additional zero.

Counting them up gives us 10 + 10 + 1 + 3 = 24 trailing zeroes.

I could only test this on my calculator up to 25! which gave 6 trailing zeroes (One each for the numbers 5,10,15,20, and two for 25).