Question

The following observation is made about the scores obtained by 100 students in an exam:
'For each student, there exists another student in the class such that their scores are at most ten marks away.'
If the above statement is false, which one of the following statements is necessarily true?

• A. For each student, the scores of all the other students are more than 10 marks away.
• B. There exists at least one student in the class for whom the scores of all the other students are more than 10 marks away.
• C. There is exactly one student in the class for whom the scores of some students are more than 10 marks away.
• D. For each student, the score of exactly one other student is more than 10 marks away.

Hint:

Negating "for every student there is a close one" flips the quantifiers: it becomes "some student has no close one".

Answer 1

The Correct Option is B

Step 1: Restate the claim in plain terms.
The original claim says every single student has at least one "buddy" somewhere in the class whose score sits within $10$ marks of their own.

Step 2: Think about what makes this claim false.
The claim fails only if we can point to at least one student who has no such buddy at all, meaning every other student's score is more than $10$ marks away from this particular student's score. We do not need this to happen for every student, just for one.

Step 3: Rule out the wrong quantities.
Option (A) wrongly demands that this "no buddy" situation happens for every student in the class, which is a much stronger claim than what a single failure requires.
Option (C) wrongly limits it to exactly one student and softens "all other students" down to just "some students," which is weaker than the falseness we found.
Option (D) talks about exactly one distant student per person, which describes a totally different kind of pattern, not the breakdown of the original claim.

Step 4: Match the correct reading.
The only reading that matches "the claim breaks down for at least one student, and for that student, every other student is far away" is option (B).

Step 5: Conclude.
\[ \boxed{\text{There exists at least one student for whom the scores of all other students are more than 10 marks away.}} \]