I was on Math.Stack Exchange the other day and I found a question that said:

How many zeroes are there in 100! ?

I quickly factored it out and said that there were 24 zeroes. However, that are only the trailing zeroes, as the person who asked the question quickly pointed out. As the days passed no one answered the question. My question is the following: is there a method to figure it out without having to compute the whole answer? I initially thought that it could be solved using many divisibility properties, but I didn't figure anything out.