Question

The statements (S1), (S2), and (S3) pertain to the scores obtained by students in an exam. The maximum possible marks in the exam is 150.
(S1) The highest score is 100.
(S2) The fourth highest score is 76.
(S3) There are at least four students whose scores are within 25 of each other.
Which one of the following options is necessarily correct?

• A. (S1) and (S2) together imply (S3)
• B. (S1) and (S3) together imply (S2)
• C. (S2) and (S3) together imply (S1)
• D. (S1) implies (S3)

Hint:

Work out how close together the top four scores must be once you fix the 1st and 4th highest values.

Answer 1

The Correct Option is A

Step 1: Understand what "within 25 of each other" means.
A group of students has scores "within 25 of each other" when the difference between the highest and the lowest score in that group is at most 25. We need to check which combination of statements forces such a group of at least four students to exist.

Step 2: Test option (A), combining (S1) and (S2).
If the highest score is \(100\) and the fourth highest score is \(76\), then the top four scores (1st, 2nd, 3rd, and 4th highest) all lie between \(76\) and \(100\).
The 2nd and 3rd highest scores must lie between the 1st and 4th, so they also fall in this same range.

Step 3: Compute the spread of these four scores.
\[ 100-76=24 \]
Since all four scores lie within a band of width \(24\), which is less than \(25\), every pair among these four students has a score difference of at most \(24\).
So these four students automatically satisfy (S3): there are at least four students whose scores are within 25 of each other.
Hence (S1) and (S2) together always force (S3) to be true, no matter what the other students score.

Step 4: Test option (B), combining (S1) and (S3).
Knowing only that the highest score is \(100\) and that some four students are within 25 of each other does not fix the fourth highest score at \(76\). The four clustered students could just as well have scores like \(40, 45, 50, 55\), which has nothing to do with the fourth highest mark. So (S2) does not have to follow.

Step 5: Test option (C), combining (S2) and (S3).
Knowing the fourth highest score is \(76\) and that some four students are close together does not force the highest score to be exactly \(100\). The highest score could be \(76\) itself (if the top four are tied) or any value above \(76\), so (S1) is not guaranteed.

Step 6: Test option (D), using (S1) alone.
Knowing only that the highest score is \(100\) tells us nothing about how the other 99 students are spread out. They could all be spaced more than 25 marks apart from each other and from the topper, so (S3) does not have to follow from (S1) alone.

Step 7: Final conclusion.
Only the combination in option (A) is forced to be true in every possible case.
\[ \boxed{\text{(S1) and (S2) together imply (S3)}} \]