Originally Posted by
PaulRS Note that in the set $\displaystyle
S_n = \left\{ {1,2,...,n} \right\}
$ there are $\displaystyle
\left\lfloor {\tfrac{n}
{p}} \right\rfloor
$ multiples of $\displaystyle p$. But $\displaystyle
\left\lfloor {\tfrac{n}
{{p^2 }}} \right\rfloor
$ of these, are multiple of $\displaystyle
p^2
$ and so on.
You must count once the numbers which are multiples of p but not of p², twice the numbers which are multiple of p² and not of p³,... , in order to get the maximum power of p dividing n!
And to do so it's enough to sum $\displaystyle
\left\lfloor {\tfrac{n}
{p}} \right\rfloor + \left\lfloor {\tfrac{n}
{{p^2 }}} \right\rfloor + ...
$