# How many combinations of three dice given as a product an even number

I want to know how many combinations of three dice given as a product an even number. If these dice are different from each other the answer would be $$6^3- 3^3$$. What about if these dice are equal to one another? Like having $$2, 4, 6$$ is the same as $$4, 6, 2$$.

• Do you know how to enumerate the number of possible results of three dice under the constraint? I.e., what takes the role of '$6^3$' if you're talking about the set of dice and not the individual dice? – Steven Stadnicki Aug 19 '19 at 22:12
• @StevenStadnicki could you be more specific? – user42912 Aug 19 '19 at 22:23

Have you heard of the stars and bars method? In this case, each die would be represented by a star. Any dice before the first bar is a 1, any between the first and second bar is a 2, etc. So, for example, |*||**|| would represent one 2 and two 4's. There are 8 possible locations to choose 3 stars, so the total number of results from rolling 3 dice is $$\binom83=56$$.
There are $$6$$ ways if all three numbers are the same. If two of the numbers are the same, and the other is different, there are $$6\cdot5=30$$ ways. If all numbers are different, there are $${6\choose3}=20$$ ways. Altogether that makes $$56$$ ways.