TITLE: How many distinguishable outcomes from rolling 6 identical dice?
QUESTION [8 upvotes]: Ignoring order, how many distinguishable outcomes are there from rolling 6 identical dice? Answer = $462$
I tried a variety of ways such as $\frac{6^6}{6!}$ and can't seem to get the answer. Struggling how to incorporate no order and distinguishable at the same time. Please help.

REPLY [2 votes]: 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:
$${6+6-1\choose 6-1}=462.$$