Concept:
Families such as Fabaceae and Solanaceae possess characteristic floral features, fruit types and pollination mechanisms. Questions based on family characteristics require careful comparison of morphological traits.
Step 1: Examine Statement I.
Diadelphous stamens are characteristic of many members of Fabaceae such as Pisum and Bean.
However, Crotalaria exhibits monadelphous stamens rather than typical diadelphous condition.
Therefore the statement that both Arachis and Crotalaria show diadelphous stamens is incorrect.
Hence Statement I is false.
Step 2: Examine Statement II.
Solanum bears berry fruits.
However, Nicotiana produces a capsule and not a berry.
Therefore Statement II is false.
Step 3: Examine Statement III.
Pisum (pea) and Lathyrus possess bisexual flowers and commonly exhibit self-pollination.
Self-pollination occurs before flowers open in many species.
Therefore Statement III is correct.
Step 4: Examine Statement IV.
In Solanaceae, the ovary is bicarpellary and syncarpous.
The carpels are not characteristically arranged obliquely at \(90^\circ\).
Hence Statement IV is false.
Step 5: Final conclusion.
Only Statement III is correct.
Therefore the correct answer is:
\[
\boxed{\text{III only}}
\]