let $m$ be a natural number, does always exist a $N\in \mathbb{N}$ such that $m$ or more "$0$" digits (excluding the terminal ones) appears amongs the decimal digits of $n!$ if $n\ge N$?
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$\begingroup$ It may be possible to show that for every m, the set of n which have fewer than m occurrences of a particular digit among, say, the leading 75% digits of n! has density 0, but I doubt that the same set is finite, much less whether it can be proved finite. Gerhard "Ask Me About System Design" Paseman, 2014.03.14 $\endgroup$– Gerhard PasemanCommented Mar 14, 2014 at 18:17
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$\begingroup$ @Gerhard Why do you doubt it's finite? $\endgroup$– user25199Commented Mar 14, 2014 at 18:37
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$\begingroup$ For the same reason that if you pick a random integer, you can find a prime that does not divide that integer, except in one case. Even though it is highly unlikely that there are infinitely many factorials with fewer than 23 8 digits in their base 10 expansion, it is possible, and my intuition does not rule out the possibility. Gerhard "Why Not Believe The Unlikely?" Paseman, 2014.03.14 $\endgroup$– Gerhard PasemanCommented Mar 14, 2014 at 18:50
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2$\begingroup$ If we model the non-terminal digits of $n!$ as a random number of the same size (which should be accurate enough for our purposes), then the probability that none of the digits equals $0$ is like $c^{n\log n}$ for some $c<1$. The sum of this quantity over $n\in\Bbb N$ converges (incredibly fast), and so Borel-Cantelli suggests that there are only finitely many $n$ for which $n!$ has no non-terminal zeros. A slight elaboration gives the same heuristic that there are only finitely many $n$ for which $n!$ has fewer than $m$ non-terminal zeros, for any $m\in\Bbb N$. $\endgroup$– Greg MartinCommented Mar 14, 2014 at 22:43
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Your question is partly answered in this question. All possible leading sequences of digits appear in $n!$, including those with $m$ zeros, so there exist infinitely many $n$. But I think proving it for all $n>N$ for some $N$ is beyond reach of current techniques.
