In a cricket team, the average age of eleven players is 28 years. What is the age of the captain of the team? Use the following additional statements: (a) The captain is eleven years older than the youngest player. (b) The average age of the other 10 players, excluding the captain, is 27.3 years. (c) Excluding 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. Which option correctly describes which statements are sufficient to determine the captain's age?

Introduction / Context:
This is a classic data sufficiency problem based on averages. The main question is the age of the captain of a cricket team, given that the average age of eleven players is known. Several extra statements provide different kinds of average information. Our task is not to calculate the age numerically for each option, but to decide which statements provide enough information to determine that age uniquely.

Given Data / Assumptions:

• Average age of 11 players is 28 years.
• Therefore, total age of all 11 players together is 11 * 28 = 308 years.
• Statement (a): Captain is 11 years older than the youngest player.
• Statement (b): Average age of the 10 players other than the captain is 27.3 years.
• Statement (c): Leaving aside the captain and the youngest player, there are 9 players divided into three groups of three with average ages 25, 28 and 30 years.

Concept / Approach:
To determine if a set of statements is sufficient, we check whether they allow us to form enough independent equations to get a unique numerical value for the captain's age. If exactly one age fits, the information is sufficient. If multiple values are possible, the information is not sufficient. If different combinations of statements each give a unique answer, then either set is sufficient.

Step-by-Step Solution:
Step 1: Use the base information. Total age of the team is 308 years.Step 2: Consider statement (b) alone. Average age of the other 10 players is 27.3 years, so their total age is 10 * 27.3 = 273 years.Step 3: Subtract from the team total. Captain's age = 308 - 273 = 35 years. Hence statement (b) alone is sufficient to determine the captain's age exactly.Step 4: Now consider statements (a) and (c) together, without using statement (b). Let the youngest player have age y. Then, from statement (a), the captain's age is y + 11.Step 5: From statement (c), the nine players excluding captain and youngest have total age 3 * (25 + 28 + 30) = 3 * 83 = 249 years.Step 6: Total age of all 11 players is then y (youngest) + (y + 11) (captain) + 249 = 2y + 260.Step 7: Equate this to the known total 308: 2y + 260 = 308, so 2y = 48 and y = 24. Therefore, captain's age = 24 + 11 = 35 years, uniquely determined again.Step 8: Thus, statement (b) alone is sufficient, and the combination of statements (a) and (c) is also sufficient to find the captain's age.

Verification / Alternative check:
Both methods lead to the same captain's age, which confirms internal consistency.Any different numbers for the youngest and captain that still satisfy the averages of the three groups in statement (c) would alter the total, but the total is fixed at 308 years.Therefore, once the structure of the equations is set up, only one age for the captain is possible in each valid method, so sufficiency is established.

Why Other Options Are Wrong:
Option a underestimates the power of statements (a) and (c); they are together sufficient even without statement (b).Option b ignores the fact that statement (b) alone already gives a unique answer.Option d is wrong because we have more than enough information; in fact, we have two independent ways of solving.Option e is incomplete because it omits the possibility of sufficiency using statement (b) alone and the pair (a) plus (c).

Common Pitfalls:
Confusing the task of data sufficiency with simply calculating the answer once, instead of checking all combinations.Forgetting to convert averages into totals before forming equations.Not recognising that multiple independent combinations of statements can each be sufficient.

Final Answer:
The correct description is that Either statement (b) alone or the combination of statements (a) and (c) is sufficient to determine the captain's age uniquely.