Combinatorics: The Step-by-Step Method

Ed throws a blue die and a red die. Both dice have 6 sides that are each numbered with one of the integers from 1 through 6. How many combinations are there in which the 2 dice do not yield the same result and the sum of the results is not 9?

Correct. [[snippet]] There aren't that many forbidden choices, so you can just count them. 1) Start with the results in which the dice are the same: > 1-1, 2-2, 3-3, 4-4, 5-5, and 6-6 $$\rightarrow$$ a total of 6 forbidden combinations where the dice yield the same result. 2) Now count results that sum up to 9: > 3-6, 6-3, 4-5, and 5-4 $$\rightarrow$$ a total of 4 forbidden combinations where the sum is 9. Since none of these cases are being double counted, there are $$6+4 = 10$$ forbidden choices. Now, using __SeBoxes__, calculate the total number of possible results for rolling 2 dice. With 6 choices for the Die 1 _and_ 6 choices for Die 2, the total number of combinations is >$$\mathop{\fbox{${6}$}}\limits^{{\text{Die 1}}}\hspace{0.33em}\mathrm{\times}\hspace{0.33em}\mathop{\fbox{${6}$}}\limits^{{\text{Die 2}}} = 36$$. So the number of good choices is > $$\text{Total} - \text{Forbidden} = 36-10 = 26 \text{ combinations}$$.

Incorrect. [[snippet]]

10

24

26

30

36