Step 1: Understanding the Concept:
This question tests the "Sequence of Tenses" in the past.
Specifically, it focuses on the relationship between the Simple Past and the Past Perfect tense.
When two actions occur in the past at different times, we use different tenses to clarify which action happened first.
Step 2: Detailed Explanation:
Let's analyze the timeline of events described in the sentence:
Action A: We reached the station.
Action B: The train left.
1. The Sequence Rule:
- The action that happens later is expressed in the Simple Past ($V2$). In this sentence, that is "reached."
- The action that happened earlier (the first action to finish) must be expressed in the Past Perfect ($had + V3$).
In this context, the train left {before} the people arrived, which is why the phrase "By the time" is used.
2. Evaluating the Options:
- Option (A) left: Using Simple Past for both actions implies they happened at roughly the same time or as a direct sequence. However, "By the time" specifically signals that the first action was already completed.
- Option (B) has left: This is Present Perfect. You cannot use a present tense auxiliary ("has") when the main frame of the sentence is already in the past ("reached"). This is a tense mismatch.
- Option (C) had left: This is Past Perfect. It correctly indicates that the train's departure was the "past of the past." This is the standard grammatical construction for this sentence type.
- Option (D) leaves: This is Simple Present. It is logically and grammatically impossible for the train to "leave" (present/habitual) "by the time we reached" (past).
Syntactic Formula:
$By\ the\ time + Subject + V2\ (Simple\ Past),\ Subject + had + V3\ (Past\ Perfect).$
Example: "By the time the police arrived, the thief had escaped."
Step 3: Final Answer:
The logic of the sentence requires the earlier action to be in the Past Perfect tense. Therefore, "had left" (Option C) is the correct choice.