Out of all the patients in a hospital 89% are found to be suffering from heart ailment and 98% are suffering from lungs infection. If K% of them are suffering from both ailments, then K can not belong to the set :

<p>This solution begins by applying the principle of inclusion and exclusion, which in the context of this problem, is represented by the formula : </p> <p>n(A ∪ B) ≥ n(A) + n(B) - n(A ∩ B)</p> <p>Here, n(A ∪ B) represents the total number of patients in the hospital, which is 100%. n(A) represents the proportion of patients with a heart ailment (89%), and n(B) represents the proportion of patients with a lung infection (98%). </p> <p>By rearranging this formula, the solution establishes an inequality for n(A ∩ B), the proportion of patients suffering from both ailments :</p> <p>100% ≥ 89% + 98% - n(A ∩ B)</p> <p>Therefore, </p> <p>n(A ∩ B) ≥ 87%</p> <p>Next, the solution notes that n(A ∩ B) cannot be greater than the smaller of n(A) and n(B), since it cannot be larger than the smallest group. Thus, we have another inequality :</p> <p>n(A ∩ B) ≤ 89%</p> <p>Combining these two inequalities gives :</p> <p>87% ≤ n(A ∩ B) ≤ 89%</p> <p>Hence, the proportion of patients suffering from both ailments must be a value between 87% and 89% inclusive. So, the set of values {79,81,83,85} which are all less than 87% are values that n(A ∩ B) cannot belong to. Therefore, option C is the correct answer.</p>