# If four identitical dice are rolled, how many different outcomes....

If four identitical dice are rolled, how many different outcomes can be recorded?

Soln: So I am aware that the solution to this follows a bars and stars sort of procedure and I beleive the solution is C(4+6 -1, 4) = C (4+6 - 1, 5) depending if you are arranging by slots or by bars. Since I am still new to the bars and stars method I am still trying to visualize how the bars and stara would be arranged in a sample outcome. Here is where I am having trouble convincing myself.

So I am dealing with 4 rolls and each dice has 6 possible results so how would a sample outcome look? would it look like this: $$****** | ****** | ****** | ******$$

where each bar is a separator for a roll and the stara represemt the numbered result? It doesn't feel like that is how it should be considered

• Do you consider, say $(1,2,2,3)$ as the same outcome as e.g. $(1,3,2,2)$ or as different outcomes? Sorry this is not clear to me. Commented Dec 12, 2015 at 14:17
• i believe the same Commented Dec 12, 2015 at 14:18
• What is outcome for you? The sum of the 4 numbers, the 4-tuple of the 4 numbers or the different possible combinations? Any of these is related to stars and bars.
– user173262
Commented Dec 12, 2015 at 14:20
• the way I am interpreting it is as the different outcomes possible. But the 4- tuple may be something I will try after Commented Dec 12, 2015 at 14:27
• Oh, yes... my mistake. This is a star and bars problems, yes. You must "invert" the groups to represent it, i.e. put 6 stars to be divided by 4 bars where the bars can stay together in the same hole.
– user173262
Commented Dec 12, 2015 at 14:38

Hint: In this problem, bars should separate the possible values a die roll can have (so $5$ bars), and stars should be the dice ($4$ stars).
So for instance $$||*|**||*$$ corresponds to $3, 4, 4, 6$ on the dice.