Fac-finding

See the notes for information about why Lyndon likes this problem. He has also added a further related question (in green below).

If factorial $100$ ($100!$) was rewritten as the product of its prime factors how many $2$s and how many $5$s would there be?

Along the way confirm how many zeros are at the end of this large number and what the first digit to precede them is.

Published September 2003,January 2007.