On which day in January did Subhas leave for Germany? Statement I: Subhas has so far spent 10 years in Germany. Statement II: Subhas's friend Anil left for Germany on 15th February and joined Subhas 20 days after Subhas's arrival.

Introduction / Context:
This problem is a data sufficiency question about dates and timing. The question asks for the exact day in January when Subhas left for Germany. Two statements give information about how long Subhas has been in Germany and when his friend Anil joined him. Your job is not to compute an approximate number but to decide whether the given information is enough to pinpoint the exact date in January.

Given Data / Assumptions:
- The question: On which day in January did Subhas leave for Germany?
- Statement I: Subhas has so far spent 10 years in Germany.
- Statement II: Anil left for Germany on 15th February and joined Subhas 20 days after Subhas's arrival.
- There is no explicit reference year; only relative timing is described.
- The length of a year and month follows the standard calendar, but specific years are not provided.

Concept / Approach:
For data sufficiency, we check each statement separately with the common background, then consider them together if necessary. The key is whether the information allows us to determine a unique calendar date for Subhas's departure in January. A mere relationship between events, like someone arriving a certain number of days earlier or later, does not necessarily give an absolute date unless at least one fixed date is clearly anchored to Subhas's travel.

Step-by-Step Solution:
Step 1: Use statement I alone. It tells us that Subhas has spent 10 years in Germany so far. However, it does not mention the current year or when that 10 year period was measured. Without knowing the present year or the year of his departure, we cannot identify the exact date in January on which he left. Step 2: Therefore, statement I alone is not sufficient. Step 3: Use statement II alone. It says that Anil left for Germany on 15th February and joined Subhas 20 days after Subhas arrived. This tells us that Subhas must have reached Germany 20 days before Anil joined him, but the statement does not connect this directly to January or provide which year is being discussed. Step 4: Even with the fixed date of 15th February for Anil's departure, we do not know how long Anil's journey took or the exact arrival date of Anil in Germany, so we cannot count back precisely to Subhas's arrival date, let alone his departure date from the home country. Step 5: Therefore, statement II alone is also not sufficient. Step 6: Now combine statements I and II. The first gives a total period of 10 years but still no starting calendar date, while the second only relates Anil's journey to Subhas's arrival by 20 days. Even together, they do not connect Subhas's starting date in January to any fixed known year or date beyond a general period of 10 years.

Verification / Alternative check:
We can try to imagine different possible timelines. For example, if Subhas left on 10th January of some year, he might still have spent 10 years in Germany by now depending on the current date. If he left on 25th January of some year, the 10 year period can also hold with a suitable choice of current date. Statement II adjusts Anil's journey relative to Subhas but does not fix Subhas's departure. Because many different January dates are consistent with both statements, they are not sufficient to answer the question uniquely.

Why Other Options Are Wrong:
- Option a is wrong because statement I alone does not provide any specific date information, only a duration.
- Option b is wrong since statement II alone also fails to pin down the exact date in January.
- Option c is incorrect because neither I nor II alone is sufficient.
- Option d is incorrect because even using both statements together still leaves multiple possible January dates that fit all the conditions.

Common Pitfalls:
A frequent mistake is to assume that knowing a total period like 10 years automatically allows you to backtrack to a unique starting date, but this requires knowing exactly when that period is measured from. Another pitfall is to over-interpret relational information such as "20 days after" as if it determines an absolute date, when in fact it only relates two events. Careful reading is essential in data sufficiency questions, and you must resist the urge to infer extra information that is not explicitly stated.

Final Answer:
Even when both statements are used together, the exact January date of Subhas's departure cannot be determined uniquely. Therefore, the correct data sufficiency choice is option E, meaning the data in both statements together are still not sufficient.