combinatorics - What is the probability to get each side at least once on $n$ throws of fair dice?

Let's say I have $n$ fair (six-sided) dice ($n>=6$) and I want to figure out the probability of getting each side at least once. I figured out a formula through elementary combinatorial reasoning: $$\frac{6!\cdot\binom{n}{6}\cdot6^{n-6}}{6^n}$$

First, of all, is this formula correct?

Second, somewhere I read that I could do this with the sieve formula. How do I use the sieve formula for this one?