To determine which option includes a rational number and an irrational number between \(6 - \sqrt{}\) and \(8 - \sqrt{}\), we need to analyze the given options.
- Option 1: \(2.6\bar{6}\) (which is rational) and \(\sqrt{7}\) (which is irrational).
- Option 2: \(2.5\) (which is rational) and \(\pi\) (which is irrational).
- Option 3: \(2.7\) (which is rational) and \(\sqrt{9}\) (which is rational, since \(\sqrt{9} = 3\)).
Now let's check the range of values:
- The number \(6\) can be approximated as roughly \(6\), and \(\sqrt{7} \approx 2.645751... \rightarrow 6 - \sqrt{7}\) is a rational number and certainly less than \(6\).
- The number \(\sqrt{8} \approx 2.82843...\) , hence \(8 - \sqrt{8}\) is also a rational number that is relatively more than \(5\).
Looking at the intervals we get:
- \(6 - \sqrt{7} \approx 6 - 2.645751 = 3.354249\)
- \(8 - \sqrt{8} \approx 8 - 2.82843 = 5.17157\)
So, we're looking for a rational number and an irrational number between approximately \(3.354\) and \(5.171\).
From the options provided:
- For option 1: \(2.6\) is outside the range and doesn't meet the requirement.
- For option 2: \(2.5\) is also outside the range.
- For option 3: \(2.7\) is also outside.
Conclusion: None of the options are suitable because they do not include both a rational and an irrational number in the defined range.
It could be a consideration to verify if the problem statements are meant otherwise, or if there's any inherent misunderstanding with sqrt assumptions, however, with given values the task bears no solution from provided problems.
If needing the derived nature please refer to alternative or see if approximation is undertaken correctly.