Step 1: Understanding the Question.
The sentence has three blanks, each needing a preposition: 'My friend and I parted __ the door __ the cabin that I had rented __ the night.' We need to pick the set of three prepositions that fits naturally at each blank.
Step 2: Key Formula or Approach.
Fill each blank one at a time using the standard preposition that goes with the phrase around it: a location point uses 'at', a possessive or 'belonging to' relationship uses 'of', and a stretch of time being rented for uses 'for'.
Step 3: Detailed Explanation.
First blank, 'parted __ the door': we say someone parts (says goodbye and separates) 'at' a specific point or place, so 'parted at the door' is the natural phrase.
Second blank, 'the door __ the cabin': this describes the door belonging to the cabin, and 'belonging to' is expressed with 'of', giving 'the door of the cabin'.
Third blank, 'rented __ the night': renting something is for a length of time, and 'for' is the preposition used to state a duration or purpose, giving 'rented for the night'.
Putting the three together: 'My friend and I parted at the door of the cabin that I had rented for the night.' This reads naturally and each preposition matches its usual collocation.
Step 4: Why the other options are wrong.
Option (B), 'for; at; of', would give 'parted for the door', which does not make sense since 'for' does not mark a location where someone parts.
Option (C), 'of; for; in', would give 'parted of the door', which is not a standard collocation, 'parted of' is not correct usage.
Option (D), 'in; of; for', would give 'parted in the door', which is unnatural since 'in the door' describes being inside the door frame, not the point of separation.
Step 5: Final Answer.
The correct sequence is 'at; of; for', matching option (A).