District size comparison – In five districts: Akbarpur smaller than Fatehpur; Dhanbad bigger than Palamu; Bara Banki bigger than Fatehpur but smaller than Palamu. Which district is the biggest?

Introduction / Context:
Translate comparative statements into a partial order and identify the maximal element. Watch for chains that force one element above all others.

Given Data / Assumptions:

• Akbarpur < Fatehpur.
• Dhanbad > Palamu.
• Bara Banki > Fatehpur but < Palamu.

Concept / Approach:
From Bara Banki > Fatehpur and Bara Banki < Palamu, we have Fatehpur < Bara Banki < Palamu. Since Dhanbad > Palamu, Dhanbad exceeds all three in that chain. Also, Akbarpur < Fatehpur, so Dhanbad is higher than Akbarpur as well.

Step-by-Step Solution:
Chain: Akbarpur < Fatehpur < Bara Banki < Palamu.Given Dhanbad > Palamu → Dhanbad is above every district in the chain.

Verification / Alternative check:
No other statement contradicts Dhanbad’s superiority; hence Dhanbad is uniquely the largest.

Why Other Options Are Wrong:
Each of Akbarpur, Fatehpur, Bara Banki, and Palamu is strictly below another district; none can be maximal.

Common Pitfalls:
Misreading “bigger than” for two different pairs; ignoring the transitive implications.

Final Answer:
Dhanbad