Twenty-seven persons attended a party. Which one of the following statements can never be true?

This is a classic graph theory (handshaking / acquaintance) logic problem.

There are 27 people. Each person can have between 0 to 26 acquaintances.

Key idea:

If each person has a different number of acquaintances, then the possible degrees must be:

0, 1, 2, 3, …, 26 (27 values)

But this creates a contradiction:

• If someone has 26 acquaintances, they know everyone else.
• Then no one can have 0 acquaintances (because that 26-acquaintance person knows everyone).
• So 0 and 26 cannot coexist together.

That means you cannot assign 27 distinct different values from 0–26 consistently.

Evaluate options:

A. One person knows all 26 others → possible
B. Everyone has different number of acquaintances →  impossible (pigeonhole contradiction)
C. Someone has odd number of acquaintances → possible
D. No triangle of mutual acquaintances → possible in graph theory
E. None → incorrect because B is impossible

Final Answer:

B. Each person in the party has a different number of acquaintances