A group of \(40\) students appeared in an examination of \(3\) subjects - Mathematics, Physics  Chemistry. It was found that all students passed in at least one of the subjects, \(20\) students passed in Mathematics, \(25\) students passed in Physics, \(16\) students passed in Chemistry, at most \(11\) students passed in both Mathematics and Physics, at most \(15\) students passed in both Physics and Chemistry, at most \(15\) students passed in both Mathematics and Chemistry. The maximum number of students passed in all the three subjects is ........... .

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.