Probability of rolling all $6$ die faces I've been struggling with this for over an hour now and I still have no good results, the question is as follows:

What's the probability of getting all the numbers from $1$ to $6$ by rolling $10$ dice simultaneously?

Can you give any hints or solutions? This problem seems really simple but I feel like I'm blind to the solution.

Hint:
If $X_{i}$ denotes the number of dice that show face $i$ then it
equals:
$1-P\left[X_{1}=0\vee X_{2}=0\vee X_{3}=0\vee X_{4}=0\vee X_{5}=0\vee X_{6}=0\right]$
Now apply inclusion/exclusion.

edit:
It leads to:$$1-\binom{6}{1}\left(\frac{5}{6}\right)^{10}+\binom{6}{2}\left(\frac{4}{6}\right)^{10}-\binom{6}{3}\left(\frac{3}{6}\right)^{10}+\binom{6}{4}\left(\frac{2}{6}\right)^{10}-\binom{6}{5}\left(\frac{1}{6}\right)^{10}=\frac{38045
}{139968}$$
