You roll three identical cubical dice simultaneously, each of which has the numbers 1, 2, 3, 4, 5, and 6 on it.
Since there's no way to tell the dice apart, (2, 4, 5) is the same roll as (5, 2, 4). The only way a roll can be distinct from another is if at least one of the numbers rolled is different.
How many distinct rolls are there?
You should have 20 or 56 combinations depending on your criteria as follows:
[6 + 3 - 1] C 3 =8 C 3 = 56 combinations. This is with repeats allowed such as 3 "1s". If you want to exclude repeats, then you would have: 6 C 3 = 20 distinct combinations.
{1, 2, 3} | {1, 2, 4} | {1, 2, 5} | {1, 2, 6} | {1, 3, 4} | {1, 3, 5} | {1, 3, 6} | {1, 4, 5} | {1, 4, 6} | {1, 5, 6} | {2, 3, 4} | {2, 3, 5} | {2, 3, 6} | {2, 4, 5} | {2, 4, 6} | {2, 5, 6} | {3, 4, 5} | {3, 4, 6} | {3, 5, 6} | {4, 5, 6} (total: 20).
And with repeats, you will have 56 distinct combinations:
{1, 1, 1}, {1, 1, 2}, {1, 1, 3}, {1, 1, 4}, {1, 1, 5}, {1, 1, 6}, {1, 2, 2}, {1, 2, 3}, {1, 2, 4}, {1, 2, 5}, {1, 2, 6}, {1, 3, 3}, {1, 3, 4}, {1, 3, 5}, {1, 3, 6}, {1, 4, 4}, {1, 4, 5}, {1, 4, 6}, {1, 5, 5}, {1, 5, 6}, {1, 6, 6}, {2, 2, 2}, {2, 2, 3}, {2, 2, 4}, {2, 2, 5}, {2, 2, 6}, {2, 3, 3}, {2, 3, 4}, {2, 3, 5}, {2, 3, 6}, {2, 4, 4}, {2, 4, 5}, {2, 4, 6}, {2, 5, 5}, {2, 5, 6}, {2, 6, 6}, {3, 3, 3}, {3, 3, 4}, {3, 3, 5}, {3, 3, 6}, {3, 4, 4}, {3, 4, 5}, {3, 4, 6}, {3, 5, 5}, {3, 5, 6}, {3, 6, 6}, {4, 4, 4}, {4, 4, 5}, {4, 4, 6}, {4, 5, 5}, {4, 5, 6}, {4, 6, 6}, {5, 5, 5}, {5, 5, 6}, {5, 6, 6}, {6, 6, 6} = 56 .
