A Question is given followed by two Statements I and II. Consider the Question and the Statements.

Question: If the average marks in a class are 60, then what is the number of students in the class?

Statement-I : The highest marks in the class are 70 and the lowest marks are 50.

Statement-II : Exclusion of highest and lowest marks from the class does not change the average.

Which one of the following is correct in respect of the above Question and the Statements?

The correct option is: The Question cannot be answered even by using both the Statements together.

Let us analyze the problem step-by-step to understand why the question cannot be answered even by combining both statements.

Given Information in the Question:
The average marks of the class = 60.
Let $N$ be the total number of students in the class, and let $S_N$ be the sum of the marks of all $N$ students.
Using the definition of average, we have:
$\frac{S_N}{N}=60$
This gives us one equation:
$S_N=60N$
Here, both $S_N$ and $N$ are unknown variables.

Analyzing Statement-I:
The highest marks in the class are 70 and the lowest marks are 50.
This statement only gives us the values of two specific data points (the maximum and minimum marks). It does not provide any information about the total number of students $N$ or the marks of the other students. Thus, Statement-I alone is not sufficient to find $N$.

Analyzing Statement-II:
Exclusion of the highest and lowest marks from the class does not change the average.
Let the highest mark be $H$ and the lowest mark be $L$.
If we exclude these two marks, the new sum of marks becomes $S_N-(H+L)$, and the number of students becomes $N-2$.
According to Statement-II, the new average is still 60:
$\frac{S_N-(H+L)}{N-2}=60$
Substitute $S_N=60N$ into the equation:
$60N-(H+L)=60(N-2)$
Simplifying this:
$60N-(H+L)=60N-120$
Subtracting $60N$ from both sides, we get:
$H+L=120$
This statement only tells us that the sum of the highest and lowest marks is 120. It does not provide the value of $N$. Thus, Statement-II alone is not sufficient.

Combining Statement-I and Statement-II:
From Statement-I, we know $H=70$ and $L=50$.
We can verify that $H+L=70+50=120$, which is consistent with the result we derived from Statement-II.
However, even with this combined information, we still have no way to determine the value of $N$. For example:
- A class of 3 students with marks {50, 60, 70} has an average of 60, and excluding {50, 70} leaves {60} with an average of 60.
- A class of 4 students with marks {50, 60, 60, 70} also has an average of 60, and excluding {50, 70} leaves {60, 60} with an average of 60.
Since multiple values of $N$ satisfy all the given conditions, the number of students in the class cannot be uniquely determined.

Therefore, the question cannot be answered even by using both the statements together.