Three fair dice: find prob. that sum is at least 14

Three fair dice are rolled. What is the probability that the sum of the numbers shown on the three dice is at least 14?

I am thinking the best way to do this is to reverse it and find the probability of rolling a total of 15, 16, 17, and 18 and subtracting the sum of those probabilities from 1.

So am i going to have to make a huge table starting from 3 6 6 to 6 6 6, count the number of times each number appears and divide that from 216 total possibilities?

Would that work and is their a quicker way to solve this one?

You are not going to like the answer. No there is no easy way to do these kinds of dice problems. Here is the most efficient way given the power of calculators or computer algebra systems.
Expand this polynomial: \(\displaystyle \left( {x + x^2 + x^3 + x^4 + x^5 + x^6 } \right)^3\).
The add up the coefficients of \(\displaystyle x^{14} , x^{15} , x^{16} , x^{17} , x^{18}\)

Note it says at least 14. Then you should get 35.