By saying "events" I mean we don't count the permutations of same numbers. For example, if a die is rolled for 2 times only, 21 different events can occur.
We consider $\{4_{(\text{die #1})}, 4_{(\text{die #2})}\}$ and $\{4_{(\text{die #2})}, 4_{(\text{die #1})}\}$ as the same events.
Under this assumption, who many events can occur if a die is rolled for 10 times? What is the general formula for $n$ rolls?
The example with $4$ and $4$ would be considered one outcome in any case. I will assume that you would also consider $3$ on die 1 and $5$ on die 2 the same event as the other way around.
Then there are $6$ "bins" and we are interested only in the number of items in each bin.
This is a standard combinatorial problem. The answer, by a Stars and Bars argument, is $\binom{n+6-1}{6-1}$.
For details, please see a this very recent question.
