Correct Option (3):

Initially, the class has 45 students, and the boy's rank from the top is 20th.

Number of students positioned before him = 19.
Number of students positioned after him = Total students - His rank = 45 - 20 = 25.

When two additional boys joined the class, the total number of students increased to 45 + 2 = 47.

His rank "dropped by one," which implies his numerical rank from the top increased by one position. Therefore, his new rank from the top is 20 + 1 = 21st.

To determine his new rank from the end of the class:

Number of students positioned after him in the new arrangement = Total new students - His new rank from the top = 47 - 21 = 26.
His new rank from the end = (Number of students positioned after him) + 1 = 26 + 1 = 27th.

Incorrect Options:

Options 1, 2, and 4 are incorrect as they result from miscalculations or an incorrect interpretation of the problem statement. Common errors could include:

Failing to update the total number of students after the two new boys joined.
Incorrectly interpreting "rank dropped by one" (e.g., assuming his rank improved to 19th, or remained 20th).
Incorrectly calculating the rank from the end (Rank from end = Total students - Rank from top + 1). For instance, if his rank from the end was calculated based on the initial total students (45) and his initial rank (20th), it would be (45 - 20) + 1 = 26th, which is his initial rank from the end and does not account for the changes.
Calculating based on 47 students but retaining his original 20th rank would yield (47 - 20) + 1 = 28th (Option 4), which is incorrect because his rank also changed.