Probability that the sum of three dice is not greater than 9 
What is the probability that the sum of 3 indistinguishable dice is less than or equal to 9?

I have tried counting the pairs $X_1, X_2$ such that their sum is less than or equal to 8, but I seem to be overcounting because I get that for each pair of dice I have 46 possibilities for their sum to be less than 8.

Consider the sum of the first two dice. If the sum is lower than or equal to 3, the third die can have any value. As the sum increases from 3 to 8, fewer possible values can be thrown by the third die. Since the number of ways to throw a sum of $n$ with two dice equals $6 - |n - 7|$, the number of ways to throw a total of 9 or less equals:
$$1 \cdot 6 + \sum_{i = 3}^{8} (6 - |i - 7|) (9 - i) = 1 \cdot 6 + 2 \cdot 6 + 3 \cdot 5 + 4 \cdot 4 + 5 \cdot 3 + 6 \cdot 2 + 5 \cdot 1 = 81$$
Since there are $6^3 = 216$ ways to throw the three dice, the probability of throwing 9 or less equals:
$$\frac{81}{216} = 0.375$$
