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.

Think of creating a string of length $6n$, where each letter is one of the numbers on the face of the die. For instance, for $n=1$, we could have the string $351264$. There are $6^{6n}$ such strings, as can be seen by elementary counting methods. The strings you are interested in are ones in which each letter appears $n$ times. Again, from combinatorial results, there are 
$$
\frac{(6n)!}{(n!)^6}
$$
such strings. To get the probability, now you just have to divide the two, as each string is equiprobable. 
This result can be easily generalized to dice with $m$ faces. 
