Official solution
This solution begins by applying the principle of inclusion and exclusion, which in the context of this problem, is represented by the formula :
n(A \cup B) \ge n(A) + n(B) - n(A \cap B)
Here, n(A \cup 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\%).
By rearranging this formula, the solution establishes an inequality for n(A \cap B), the proportion of patients suffering from both ailments :
100\% \ge 89\% + 98\% - n(A \cap B)
Therefore,
n(A \cap B) \ge 87\%
Next, the solution notes that n(A \cap 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 :
n(A \cap B) \le 89\%
Combining these two inequalities gives :
87\% \le n(A \cap B) \le 89\%
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 \cap B) cannot belong to. Therefore, option C is the correct answer.