jmc

There are $\left(\genfrac{}{}{0ex}{}{7}{4}\right)=35$ ways to choose 4 of the islands. For each choice, there is a probability of ${\left(\frac{1}{5}\right)}^4{\left(\frac{4}{5}\right)}^3$ that those 4 islands will have treasure and the others will not. Therefore, the total probability that exactly 4 of the islands have treasure is $35{\left(\frac{1}{5}\right)}^4{\left(\frac{4}{5}\right)}^3=\boxed{\frac{448}{15625}}$.