Step 1: What controls the boiling point of an alcohol.
Alcohols boil the way they do mostly because of two things pulling in opposite directions: a longer, heavier carbon chain gives stronger van der Waals attraction between molecules and raises the boiling point, while branching near the $-OH$ makes the molecule more compact and rounded, which weakens those same attractions and lowers the boiling point.
Step 2: Sorting the compounds by chain length first.
Propan-1-ol has three carbons, butan-1-ol and butan-2-ol both have four, and pentan-1-ol has five, so purely by mass we already expect propan-1-ol to boil lowest and pentan-1-ol to boil highest, with the two butanols sitting in between.
Step 3: Breaking the tie between the two four-carbon alcohols.
Butan-1-ol is a straight primary alcohol, while butan-2-ol carries its $-OH$ on the second carbon, making the molecule slightly more compact in shape. That extra compactness lowers the intermolecular attraction a bit, so butan-2-ol boils just below butan-1-ol.
Step 4: Putting the full order together.
Stringing all of this together gives propan-1-ol lowest, then butan-2-ol, then butan-1-ol, then pentan-1-ol highest, which matches the increasing order given as the correct choice. \[ \boxed{\text{Propan-1-ol} < \text{Butan-2-ol} < \text{Butan-1-ol} < \text{Pentan-1-ol}} \]