Question:
Twenty-seven persons attend a party. Which one of the following statements can never be true?
Twenty-seven persons attend a party. Which one of the following statements can never be true?
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A person at this party can know anywhere from 0 to 26 others. Ask what would have to happen if all 27 counts had to be different.
Updated On: Nov 7, 2026
- There is a person in the party who is acquainted with all the twenty-six others.
- Each person in the party has a different number of acquaintances.
- There is a person in the party who has an odd number of acquaintances.
- In the party, there is no set of three mutual acquaintances.
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The Correct Option is B
Solution and Explanation
Step 1: Understand what is asked.
We have 27 people. A statement "can never be true" means no possible arrangement of acquaintances at the party can make it true. The three other statements can each be made true in at least one arrangement. Acquaintance is mutual: if A knows B, then B knows A.
Step 2: Check statement 1: someone knows all 26 others.
Let one person, say P, know everybody, and let nobody else know anybody except P. Then P knows all 26 others, so the statement is true in this arrangement. So option 1 can be true.
Step 3: Check statement 2: everybody has a different number of acquaintances.
A person in a party of 27 can know between 0 and 26 others, so there are exactly 27 possible values: 0, 1, 2, ..., 26.
If all 27 people have different counts, every one of these values must be used exactly once. So someone knows nobody (count 0) and someone knows all 26 others (count 26).
But the person who knows all 26 others must know the person who knows nobody. Then that person knows at least one other person, so cannot have count 0. This is a contradiction.
So statement 2 can never be true.
Step 4: Check statement 3: someone has an odd number of acquaintances.
Take the arrangement where only two people, A and B, know each other and nobody else knows anyone. Then A has 1 acquaintance, which is odd. So option 3 can be true.
Step 5: Check statement 4: no set of three mutual acquaintances.
If nobody knows anyone, there are no acquaintances at all, so there is certainly no group of three who all know each other. So option 4 can be true.
Final Answer:
Only statement 2 is impossible, because the counts 0 and 26 cannot both appear in the same party. This is option 2. \[ \boxed{2} \]
We have 27 people. A statement "can never be true" means no possible arrangement of acquaintances at the party can make it true. The three other statements can each be made true in at least one arrangement. Acquaintance is mutual: if A knows B, then B knows A.
Step 2: Check statement 1: someone knows all 26 others.
Let one person, say P, know everybody, and let nobody else know anybody except P. Then P knows all 26 others, so the statement is true in this arrangement. So option 1 can be true.
Step 3: Check statement 2: everybody has a different number of acquaintances.
A person in a party of 27 can know between 0 and 26 others, so there are exactly 27 possible values: 0, 1, 2, ..., 26.
If all 27 people have different counts, every one of these values must be used exactly once. So someone knows nobody (count 0) and someone knows all 26 others (count 26).
But the person who knows all 26 others must know the person who knows nobody. Then that person knows at least one other person, so cannot have count 0. This is a contradiction.
So statement 2 can never be true.
Step 4: Check statement 3: someone has an odd number of acquaintances.
Take the arrangement where only two people, A and B, know each other and nobody else knows anyone. Then A has 1 acquaintance, which is odd. So option 3 can be true.
Step 5: Check statement 4: no set of three mutual acquaintances.
If nobody knows anyone, there are no acquaintances at all, so there is certainly no group of three who all know each other. So option 4 can be true.
Final Answer:
Only statement 2 is impossible, because the counts 0 and 26 cannot both appear in the same party. This is option 2. \[ \boxed{2} \]
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