Throwing 2 labeled dice how many outcomes have sum divisible

Throwing 2 labeled dice, how many outcomes have sum divisible by 3? How many outcomes have sum at least 9?

Solution

The sums can range from 2 (1 + 1) to 12 (6+6).

Thus, those divisible by 3 are: 3, 6, 9, 12.

Trying to enumerate the number of ways and as we count, we know that

For 3: 2 ways
For 6: 5 ways
For 9: 4 ways
For 12: 1 way

Thus, there are a total of 12 ways to get a divisible by 3 result. [ANSWER, 12]

**************************

For at least 9:

For 9: 4 ways
For 10: 3 ways
For 11: 2 ways
For 12: 1 way

Thus, there are a total of 10 ways to get an \"at least 9\" result. [ANSWER, 10]