Step 1: Picture the line of students.
Imagine all students standing in a row by rank. One student is 35th counting from the top and 32nd counting from the bottom.
Step 2: Count those above the student.
Being 35th from the top means there are $35 - 1 = 34$ students ahead of this student.
Step 3: Count those below the student.
Being 32nd from the bottom means there are $32 - 1 = 31$ students behind this student.
Step 4: Add the two groups plus the student.
Total $= 34 \text{ above} + 31 \text{ below} + 1 \text{ (the student)} = 66$.
Step 5: Check with the standard formula.
$\text{Total} = \text{rank from top} + \text{rank from bottom} - 1 = 35 + 32 - 1 = 66$, since the student is counted once in each ranking and we remove the double count.
Step 6: Conclude.
Both methods agree, so the class has 66 students, matching option 1.
\[ \boxed{66} \]