Rohan went to the post-office to buy five-rupee, two-rupee and one-rupee stamps. He paid the clerk ₹ $20$, and since he had no change, he gave Rohan three more one-rupee stamps. If the number of stamps of each type that he had ordered initially was more than one, what was the total number of stamps that he bought?

Concept & Logic This is a **Linear Diophantine Equation** problem with boundary conditions. We need to form an equation for the total cost based on the number of stamps, adjust for the "change" given as extra stamps, and test integer values that satisfy the condition (more than one of each type). Step-by-Step Solution * **Understanding the Cost:** Rohan gave ₹ $20$. The clerk had no change, so instead of returning money, the clerk gave $3$ extra one-rupee stamps. This implies the clerk owed Rohan exactly ₹ $3$ in change. Therefore, the true cost of Rohan's *initial* order was ₹ $20$ - ₹ $3$ = ₹ $17$. * **Setting up the Equation:** Let the initial number of ₹ $5$ stamps be $x$. Let the initial number of ₹ $2$ stamps be $y$. Let the initial number of ₹ $1$ stamps be $z$. Total cost: $5x + 2y + z = 17$ Condition: He ordered *more than one* of each type, so $x \ge 2$, $y \ge 2$, $z \ge 2$. * **Testing Values:** Since $5x$ grows fastest, test $x$ first. If $x = 3$: Cost is $15$. Remaining for $y$ and $z$ is $17 - 15 = 2$. But $y \ge 2$ alone costs ₹ $4$. So $x$ cannot be $3$. Therefore, $x$ MUST be $2$. If $x = 2$: $5(2) + 2y + z = 17 \Rightarrow 10 + 2y + z = 17 \Rightarrow 2y + z = 7$. Now test valid values for $y$ (where $y \ge 2$): If $y = 2$: $2(2) + z = 7 \Rightarrow 4 + z = 7 \Rightarrow z = 3$. (Valid, as $z \ge 2$) If $y = 3$: $2(3) + z = 7 \Rightarrow 6 + z = 7 \Rightarrow z = 1$. (Invalid, as $z$ must be $\ge 2$) So the initial order was exactly: $x = 2$, $y = 2$, $z = 3$. Initial stamps = $2 + 2 + 3 = 7$ stamps. * **Final Calculation:** The question asks for the total number of stamps he *bought* (walked away with). He got his initial $7$ stamps, plus the $3$ extra one-rupee stamps given as change. Total = $7 + 3 = 10$ stamps. Exam Strategy & Shortcut Deduct the change logically first: he bought ₹$17$ worth of stamps. Since he bought at least $2$ of each, subtract the minimum baseline: $2 \times ₹5 + 2 \times ₹2 + 2 \times ₹1 = ₹10 + ₹4 + ₹2 = ₹16$. He has ₹$1$ left to spend. He can only spend it on one more ₹$1$ stamp. So his base order is $2, 2, 3$. That's $7$ stamps. Add the $3$ given as change to immediately get $10$. Common Pitfall Failing to read the final question carefully. Many students will correctly deduce the initial order ($7$ stamps) and look for it in the options, or they might calculate the total cost instead of the count. Always verify what the final sentence is asking for. Final Answer Therefore, the correct answer is **10**.