Correct Option (2)

Let the number of students in the class be $x$ . According to the problem statement, each student contributes an amount in rupees equal to the number of students. Therefore, the total contribution from all students, excluding any additional amount, is $x \times x = x^{2}$ rupees.

An additional contribution of 2 rupees was made by one student. The total collection is stated to be 443 rupees.

Thus, the total collection can be represented by the equation:

$x^{2} + 2 = 443$

To determine the sum contributed solely by the students based on their number, subtract the additional contribution from the total collection:

$x^{2} = 443 - 2$

$x^{2} = 441$

To find the number of students, take the square root of 441:

$x = \sqrt{441}$

$x = \pm 21$

Since the number of students cannot be a negative value, the positive root is considered.

Therefore, the number of students in the class is 21.

Incorrect Options:

The fundamental equation derived from the problem statement is $x^{2} = 441$ , where $x$ represents the number of students. Let's evaluate the other options:

If the number of students were 12 (Option 1), the total contribution based on the number of students would be $12^{2} = 144$ . Adding the additional 2 rupees, the total collection would be $144 + 2 = 146$ , which does not match the given total of 443 rupees.
If the number of students were 43 (Option 3), the total contribution based on the number of students would be $43^{2} = 1849$ . Adding the additional 2 rupees, the total collection would be $1849 + 2 = 1851$ , which does not match the given total of 443 rupees.
If the number of students were 45 (Option 4), the total contribution based on the number of students would be $45^{2} = 2025$ . Adding the additional 2 rupees, the total collection would be $2025 + 2 = 2027$ , which does not match the given total of 443 rupees.

Only $x = 21$ satisfies the condition $x^{2} = 441$ , resulting in a total collection of $21^{2} + 2 = 441 + 2 = 443$ rupees, consistent with the problem statement.