Step 1: Find the sentence that sets up a reported thought.
Q reads: "She said to herself silently at first, that." The word "that" at the end signals a subordinate clause is about to follow, so Q must be the opening fragment.
Step 2: Decide what follows "that."
R reads: "Without neglecting anything of what was due to her important mission." This works as a clause that qualifies what she thought, coming right after "that" and before the main idea of the thought.
Step 3: Identify the core of her thought.
P reads: "It would perhaps be possible for you to be of some use to this brave little boy." This is the actual content of what she silently said to herself, following the qualifying clause in R.
Step 4: Identify the closing reaction.
S reads: "And this idea pleased her." This naturally closes the passage as a reaction to the thought just described.
Step 5: Assemble and match.
Putting it together: "She said to herself silently at first, that, without neglecting anything of what was due to her important mission, it would perhaps be possible for you to be of some use to this brave little boy. And this idea pleased her." This is the order Q-R-P-S, which is option (c).