Three fair dice are rolled. What is the probability that the sum of the numbers appearing on them is $10$?

Solution

1. The total number of outcomes when rolling three dice is $6^3=216$.
2. The number of integer solutions to $x_1+x_2+x_3=10$ with $1\le x_i\le 6$ is the coefficient of $x^{10}$ in $(x+x^2+x^3+x^4+x^5+x^6{)}^3$.
3. This coefficient is $27$.
4. The probability is $27/216$.
5. Hence the answer is (B).