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?

The correct option is: (S1) and (S2) together imply (S3)

Let us analyze the given statements step-by-step to understand why (S1) and (S2) together imply (S3).

First, let the scores of the students be arranged in descending order:

$x_1\ge x_2\ge x_3\ge x_4\ge ...$

where $x_1$ represents the highest score, $x_2$ represents the second highest, $x_3$ represents the third highest, and $x_4$ represents the fourth highest score.

According to statement (S1), the highest score is 100:

$x_1=100$

According to statement (S2), the fourth highest score is 76:

$x_4=76$

Since the scores are ordered such that $x_1\ge x_2\ge x_3\ge x_4$, we can conclude that the scores of the top four students ($x_1,x_2,x_3,x_4$) must all lie within the closed interval $[76,100]$.

To find the maximum possible difference between the scores of any two students among these top four, we calculate the difference between the highest score and the lowest score in this group:

$\text{Maximum Difference}=x_1-x_4=100-76=24$

Since the maximum difference between any two scores in this group of four students is 24, the score of any student in this group is within 24 of any other student's score in the group. Because 24 is less than or equal to 25, it is guaranteed that the scores of these four students are within 25 of each other.

This directly satisfies statement (S3), which states that there are at least four students whose scores are within 25 of each other. Therefore, (S1) and (S2) together necessarily imply (S3).