Friday, December 15, 2017

What is the total number of zeroes in n!?

What is the total number of zeroes in $n!$?


I do not want to know the number of trailing zeroes in $n!$.


Let us take an example to understand what I want to know.


$7! = 5040$. The number of trailing zero in $7!$ is $1$. But the total number of zeroes in $7!$ is $2$.


I would like to know if there is any formula that gives me directly the total number of zeros in $n!$.


Can you help me derive one?

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