In a hospital, there were 200 Diabetes, 150 Hyperglycaemia and 150 Gastro-enteritis patients. Of these, 80 patients were treated for both Diabetes and Hyperglycaemia. Sixty patients were treated for Gastro-enteritis and Hyperglycaemia, while 70 were treated for Diabetes and Gastro-enteritis. Some of these patients have all the three diseases. Doctor Dennis treats patients with only Diabetes. Doctor Hormis treats patients with only Hyperglycaemia and Doctor Gerard treats patients with only Gastro-enteritis. Doctor Paul is a Generalist. Therefore, he can treat patients with multiple diseases. Patients always prefer a specialist for their disease. If Dr. Dennis had 80 patients, then the other three doctors can be arranged in terms of the number of patients treated as

### Concept & Logic To evaluate the specialists, we must find the exact "Only [Disease]" values. To evaluate the generalist, we must sum all patients with two or more diseases. We can unlock the entire Venn diagram by using Dr. Dennis ("Only Diabetes") to algebraically find the central "All three" intersection. $$\text{Only } D = D - (D \cap H) - (D \cap G) + (D \cap H \cap G)$$ ### Step-by-Step Solution * **Given:** * Total $D = 200$, Total $H = 150$, Total $G = 150$ * $D \cap H = 80$, $G \cap H = 60$, $D \cap G = 70$ * Dr. Dennis (Only $D$) = $80$ * Let $x$ be the patients with all three diseases. * **Step 1: Find $x$ (All three diseases)** * Using the formula for Only $D$: * $80 = 200 - 80 - 70 + x$ * $80 = 50 + x \Rightarrow x = 30$ * **Step 2: Calculate patients for Specialists** * Dr. Hormis (Only $H$) $= H - (D \cap H) - (G \cap H) + x$ * Dr. Hormis $= 150 - 80 - 60 + 30 = 40$ * Dr. Gerard (Only $G$) $= G - (D \cap G) - (G \cap H) + x$ * Dr. Gerard $= 150 - 70 - 60 + 30 = 50$ * **Step 3: Calculate patients for Generalist (Dr. Paul)** * Dr. Paul treats anyone with $\ge 2$ diseases. * Total multiple diseases $= (D \cap H) + (G \cap H) + (D \cap G) - 2x$ * Dr. Paul $= 80 + 60 + 70 - 2(30) = 210 - 60 = 150$ * **Deduction (Arrangement):** * Paul ($150$) > Gerard ($50$) > Hormis ($40$) ### Exam Strategy & Shortcut Once you determine the central intersection $x = 30$, immediately calculate Dr. Hormis ($40$) and Dr. Gerard ($50$). At this point, Gerard > Hormis. Look at the options: only (b) and (c) have Gerard > Hormis. By roughly estimating the overlaps ($80, 70, 60$), it's visually obvious that Dr. Paul handles a massive bulk of patients (well over $100$), guaranteeing he is first. You can confidently pick Paul > Gerard > Hormis without finishing the exact calculation for Paul. ### Common Pitfall When calculating the total number of people in the overlapping regions, a common mistake is just adding $(D \cap H) + (G \cap H) + (D \cap G)$ to get $210$. This counts the central "all three" region ($x$) three distinct times. To get the actual number of individuals, you must subtract $2x$ to ensure the center is only counted once. ### Final Answer **Therefore, the correct answer is Paul > Gerard > Hormis.**