How many distinguishable outcomes from rolling 6 identical dice?

An outcome here is the same as a six-tuple of non-negative integers that sum to $6$, the $i^{th}$ entry telling you how many times $i$ came up as a value.

Stars and Bars tell us that the number of such is $$\binom {6+6-1}6=462$$

Let $x_1,x_2,...,x_6$ indicate the number of $1,2,3,4,5,6$.

Then the problem can be formulated as: $$x_1+x_2+x_3+x_4+x_5+x_6=6, 0\le x_i\le 6.$$

For example, the following outcomes are equivalent: $$111112\equiv 111121\equiv 111211\equiv 112111\equiv 121111 \Rightarrow \\ (x_1,x_2,x_3,x_4,x_5,x_6)=(5,1,0,0,0,0);\\ 111123\equiv 111213\equiv 112113\equiv 121113\equiv \cdots\equiv 321111 \Rightarrow \\ (x_1,x_2,x_3,x_4,x_5,x_6)=(4,1,1,0,0,0);\\$$

Using Stars and Bars method: $${6+6-1\choose 6-1}=462.$$