Step 1: Understanding the sentences.
We are asked to fill in the blanks with the correct prepositions from the given options. Let's go through each sentence:
1. "The painting is hanging the wall." The appropriate preposition here is "on," as we say something is "hanging on the wall."
2. "On hearing the fire alarm, she ran of the room." The correct preposition here is "out," as she ran "out of the room."
3. "He threw the letter the fire." The proper preposition here is "into," as we say something is thrown "into the fire."
Step 2: Analyzing the options.
- (A) into: This is correct for sentence 3.
- (B) on: This is correct for sentence 1.
- (C) across: This doesn't fit any sentence in this context.
- (D) up: This doesn't work for sentence 2.
- (E) at: This is incorrect in sentence 2.
- (F) out: Correct for sentence 2.
Step 3: Conclusion.
The correct sequence of prepositions is: \[ \boxed{B, F, A} \]
