Directions: Each of the questions given below consists of a question followed by three statements. You have to study the question and the statements and decide which of the statement(s) is/are necessary to answer the question. In a cricket eleven, the average age of eleven players is $28$ years. What is the age of the captain? I. The captain is eleven years older than the youngest player. II. The average age of 10 players, other than the captain is $27.3$ years. III. Leaving aside the captain and the youngest player, the average ages of three groups of three players each are $25$ years, $28$ years and $30$ years respectively.

### Concept & Logic This is a Data Sufficiency puzzle centered on total sums. The core principle is that the total sum of a group is its average multiplied by the number of items: $$Total Age = Average Age \times Number of People$$ To find a specific unknown (the captain's age, $C$), we need statements that either isolate $C$ directly by removing everyone else, or provide enough simultaneous equations to solve for $C$ alongside other unknowns (like the youngest player, $Y$). ### Step-by-Step Solution **Given:** * Total players = $11$ * Average age of 11 players = $28$ years. * Total age of all 11 players = $11 \times 28 = 308$ years. **Calculation / Deduction:** Let the captain's age be $C$ and the youngest player's age be $Y$. * **Evaluate Statement II alone:** It gives the average age of the 10 players excluding the captain as $27.3$ years. Total age of these 10 players = $10 \times 27.3 = 273$ years. Age of Captain ($C$) = (Total age of 11) - (Total age of 10) = $308 - 273 = 35$ years. *Statement II alone is sufficient.* * **Evaluate Statement I alone:** $C = Y + 11$. We have two variables and only one equation. We cannot find $C$. *Statement I alone is not sufficient.* * **Evaluate Statement III alone:** It leaves aside $C$ and $Y$ (removing 2 players, leaving 9). The 9 players are split into three groups of 3, with averages 25, 28, and 30. Total age of these 9 players = $(3 \times 25) + (3 \times 28) + (3 \times 30) = 75 + 84 + 90 = 249$ years. Therefore, the sum of the remaining two players ($C + Y$) = $308 - 249 = 59$. We have $C + Y = 59$. Two variables, one equation. Cannot find $C$. *Statement III alone is not sufficient.* * **Combine Statements I and III:** From III, we know $C + Y = 59$. From I, we know $C - Y = 11$. Adding both equations: $2C = 70 \Rightarrow C = 35$. *Statements I and III together are sufficient.* Thus, we can answer the question using **Statement II only**, OR by combining **Statements I and III**. ### Exam Strategy & Shortcut Scan the statements for the most direct route first. Statement II explicitly isolates the captain by providing the total for everyone else. That immediately tells you II is sufficient alone. This eliminates options (a), (b), and (d). Now you just need to check if I and III can combine. Statement I gives a difference ($C - Y$), and Statement III gives the sum of all players *except* $C$ and $Y$, which easily translates to the sum of $C + Y$. A sum and a difference of two variables will always yield their exact values. You can deduce "II only or I and III only" without doing the arithmetic ($308 - 273$ or $75+84+90$). ### Common Pitfall A major trap is assuming that because Statement II is sufficient, the answer must be "II only" (which isn't an option here, but might lead a student to guess randomly). In Data Sufficiency, you must rigorously test the other statements even after finding one that works, because combinations of the remaining, seemingly weaker statements often form a secondary valid path to the solution. ### Final Answer Therefore, the correct answer is **II only or I and III only**.