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?

Guest Jan 8, 2020

#2**0 **

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 .

Guest Jan 8, 2020

edited by
Guest
Jan 8, 2020

edited by Guest Jan 8, 2020

edited by Guest Jan 8, 2020