After 6n roll of dice, what is the probability each face was rolled exactly n times? This is closely related to the question "If you toss an even number of coins, what is the probability of 50% head and 50% tail?", but for dice with 6 possible results instead of coins (with 2 possible results). Actually I would like a more general approximation formula, for dice with m faces.

To add to the other two answers, you can use Stirling's approximation of the factorial to get a sense of the asymptotic behavior of this expression:
$$
\frac{(mn)!}{(n!)^mm^{mn}} \approx \frac{\sqrt{2\pi mn}(mn/e)^{mn}}{(2\pi n)^{m/2}(n/e)^{mn}m^{mn}}=\frac{\sqrt{m}}{(2\pi n)^{(m-1)/2}}
$$
So, for fixed $m$, the probability decays like $n^{-(m-1)/2}$; in particular, when $m=6$, it decays like $n^{-5/2}$.
