Let the total number of students in the class be denoted by $N$. We are given that:

• The number of students scoring $55$ marks or less is 24.
• The number of students scoring more than 55 marks is at most 32.
• The average score of all the students is 71 marks.

Let:

• $N_1 = 24$ be the number of students scoring 55 or less.
• $N_2 \leq 32$ be the number of students scoring more than 55 marks. So, the total number of students $N = N_1 + N_2$.

Step 1: Setting up the total score equation

Let the total score of students scoring 55 or less be $S_1$, and their average score be 55 or less. Since there are 24 students, we can denote the total score as:

$S_1 \leq 24 \times 55 = 1320$

Let the total score of students scoring more than 55 be $S_2$. The average score of all the students is 71 marks, so the total score of all students is:

$S_1 + S_2 = 71 \times N$

Thus, we have the equation:

$S_1 + S_2 = 71 \times (N_1 + N_2)$ $S_1 + S_2 = 71 \times (24 + N_2)$

Step 2: Maximizing the total score of students scoring 55 or less

To minimize the average score of students scoring more than 55 marks, we should assume that the students scoring 55 or less have the highest possible score (because the higher the score of the 55-or-less group, the lower the score required by the more-than-55 group to maintain the overall average).

Thus, the total score of students scoring 55 or less is $S_1 = 1320$.

Step 3: Calculating the score of students scoring more than 55

Substitute $S_1 = 1320$ into the equation $S_1 + S_2 = 71 \times (24 + N_2)$:

$1320 + S_2 = 71 \times (24 + N_2)$ $S_2 = 71 \times (24 + N_2) - 1320$ [ S_2 = 1704 + 71N