The probability that an island has treasure and no traps is $\frac{1}{4}$.
The probability that an island has traps but no treasure is $\frac{1}{12}$.
The probability that an island has neither traps nor treasure is $\frac{2}{3}$.
We want to find the probability that the pirate will encounter exactly 3 islands with treasure and none with traps.
This can be done by multiplying the probabilities of the individual islands.
Since there are 6 islands, we need to choose 3 of them to have treasure.
The probability of 3 islands having treasure and none having traps is $\left(\frac{1}{4}\right)^3 \left(\frac{1}{12}\right)^0 \left(\frac{2}{3}\right)^3$.
However, there are $\binom{6}{3} = 20$ ways to choose 3 islands out of 6.
So, the total probability is $20\left(\left(\frac{1}{4}\right)^3 \left(\frac{1}{12}\right)^0 \left(\frac{2}{3}\right)^3\right) = 20\left(\frac{1}{4}\right)^3 \left(\frac{2}{3}\right)^3 = 20\left(\frac{1}{64}\right)\left(\frac{8}{27}\right) = \boxed{\frac{5}{54}}$.
The answer is: \frac{5}{54}
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
A pirate is searching for buried treasure on 6 islands. On each island, there is a $\frac{1}{4}$ chance that the island has buried treasure and no traps, a $\frac{1}{12}$ chance that the island has traps but no treasure, and a $\frac{2}{3}$ chance that the island has neither traps nor treasure. What is the probability that while searching all 6 islands, the pirate will encounter exactly 3 islands with treasure, and none with traps?

### Answer:
