A fair six-faced dice, with the faces labelled ‘1’, ‘2’, ‘3’, ‘4’, ‘5’, and ‘6’, is rolled thrice. What is the probability of rolling ‘6’ exactly once?

This is a binomial probability problem.Let $n$ be the number of trials (rolls), so $n=3$.Let $p$ be the probability of success (rolling a '6' on a single die), so $p=\frac{1}{6}$.Let $q$ be the probability of failure (not rolling a '6'), so $q=1-\frac{1}{6}=\frac{5}{6}$.We want the probability of exactly $k=1$ success.Using the binomial probability formula:
$$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k{q}^{n-k}$$
$$P(X=1)=\left(\genfrac{}{}{0ex}{}{3}{1}\right){\left(\frac{1}{6}\right)}^1{\left(\frac{5}{6}\right)}^2$$
$$P(X=1)=3\times \frac{1}{6}\times \frac{25}{36}$$
$$P(X=1)=\frac{3\times 25}{6\times 36}=\frac{75}{216}$$