In a family of four A, B, C and D; D is taller than B, C is shortest of all…

Concept:

In a linear-ordering (ranking) puzzle, each comparative clue ("X is taller than Y") fixes only a relative position, not an absolute one. The reliable method is to translate every clue into a single chain of inequalities, pin the extreme (the stated shortest or tallest) first, then place the remaining people so that every count-based clue ("taller than only one person") is satisfied. A position is fully fixed only when no two people can swap without breaking a clue.

Application:

1. Clue "C is shortest of all" fixes the bottom: C occupies the lowest rank.

2. Clue "A is taller than only one person" means exactly one person is below A. Since C is already the lowest, that one person must be C, so A sits directly above C (third from the top).

3. That places A below the remaining two, B and D, so both B and D are taller than A.

4. Clue "D is taller than B" orders the top two: D above B.

Combining all four steps gives the unique height order, tallest to shortest:

D > B > A > C

Cross-check the five statements against this order:

• "Number of persons taller than A = number of persons shorter than B": taller than A = {B, D} = 2; shorter than B = {A, C} = 2; 2 = 2, so this holds.

• "D is tallest": D is at the top, so this holds.

• "A is shortest": C is the shortest, not A, so this fails.

• "No one is shorter than C": C is the lowest rank, so this holds.

• "Only one person is shorter than B": two people (A and C) are shorter than B, so this fails.

The statements that hold are the first, second and fourth, i.e. A, B and D. Hence the correct choice is "A, B and D only".