In the following question, a sentence is given with a blank space. Choose the correct phrasal verb to complete the sentence meaningfully: It is mainly due to Peter's lethargy that the plan fell _____.

Introduction / Context:
This question tests your knowledge of phrasal verbs, which are verb + particle combinations with meanings that are often idiomatic. The sentence explains that Peter's lethargy caused a plan to fail. You must choose the particle that forms a phrasal verb with fell to express the idea of failure correctly and naturally.

Given Data / Assumptions:
- Sentence: It is mainly due to Peter's lethargy that the plan fell _____.
- Options: off, through, in, out.
- The intended meaning is that the plan failed or did not work as expected because Peter was lethargic.

Concept / Approach:
The relevant concept is the phrasal verb fall through, which means to fail to happen, usually because arrangements are not completed or something goes wrong. This fits perfectly with a plan that did not succeed. The other particles combine with fall to create different meanings: fall off (decrease or drop), fall in (collapse inward or line up in formation), and fall out (argue or drop out). None of these match the idea of a plan failing to materialise.

Step-by-Step Solution:
Step 1: Identify the intended idea: the plan failed or was cancelled due to Peter's lack of energy or enthusiasm.Step 2: Consider fell off. This usually means decreased (for example, sales fell off) or physically dropped, not that a plan failed.Step 3: Consider fell through. The phrasal verb fall through is commonly used with plans, deals, or arrangements, meaning that they failed to happen.Step 4: Consider fell in. This can describe a roof falling in or soldiers forming a line but not a plan failing.Step 5: Consider fell out. Fall out often refers to quarrelling (they fell out over money) or something dropping out of place; again, not the right sense for a plan failing.Step 6: Therefore, through is the correct particle, forming fell through, which perfectly fits the context.

Verification / Alternative check:
Replace the blank with through and read the sentence: It is mainly due to Peter's lethargy that the plan fell through. This is a very common and idiomatic expression in English. If we substitute any of the other options, the sentence becomes unnatural or changes meaning: the plan fell off, fell in or fell out do not convey failed to happen in standard usage. This confirms that through is the only appropriate choice.

Why Other Options Are Wrong:
- Off suggests reduction or physical dropping, which does not describe what happens to a plan.
- In suggests an inward collapse or lining up, which is not relevant to a plan failing.
- Out suggests arguing or dropping from a group, again unrelated to the intended meaning here.

Common Pitfalls:
Phrasal verbs are tricky because their meanings cannot always be guessed from individual words. Learners may try to rely on literal translation from their first language or on guesswork with particles like off, out, in, and through. A better strategy is to learn common phrasal verbs in chunks with examples, such as plans fall through, negotiations break down, and deals fall apart. Recognising fall through as the standard expression for a plan failing will help you answer similar questions quickly in exams.

Final Answer:
The correct phrasal verb is fell through, so the sentence should read: It is mainly due to Peter's lethargy that the plan fell through.