Minimum number who can speak Hindi only (guaranteed count): In a group of 1000 people, 750 can speak Hindi and 400 can speak English. What is the minimum number who must be Hindi-only speakers?

Introduction / Context:
Without explicit overlap, exact counts of language-only groups cannot be uniquely determined. However, a minimum guaranteed number for Hindi-only can be found using inclusion-exclusion and the constraint that at most 100% of people exist.

Given Data / Assumptions:

• Total people = 1000
• Hindi speakers H = 750
• English speakers E = 400
• No statement that everyone speaks at least one language

Concept / Approach:
The size of the overlap satisfies |H ∩ E| ≥ |H| + |E| − 1000 = 150 (pigeonhole bound). Minimizing Hindi-only means maximizing the overlap.

Step-by-Step Solution:
Minimum overlap = 150Hindi-only minimum = |H| − |H ∩ E| = 750 − 150 = 600

Verification / Alternative check:
An arrangement meeting the bound exists: 150 bilingual, 600 Hindi-only, 250 English-only or outsiders adjusted accordingly.

Why Other Options Are Wrong:
650, 750, 800 exceed the guaranteed minimum; the question asks for the minimum that must be Hindi-only.

Common Pitfalls:
Subtracting 400 from 750 directly (ignores total cap) or assuming everyone speaks at least one language when not stated.

Final Answer:
600