This is a classic problem involving sets and the principle of inclusion-exclusion. Let's break it down:

Key Information:

• $|A| = 35$: Students who like going to the beach.
• $|B| = 25$: Students who like visiting the swimming pool.
• $|A \cap B| = 6$: Students who like both.
• $|A^c \cap B^c| = 3$: Students who like neither.

We are tasked to find the total number of students in the class.

Formula for Total Students:

Using the inclusion-exclusion principle for two sets:

$|A \cup B| = |A| + |B| - |A \cap B|$

The total number of students in the class includes those who like neither activity:

$\text{Total Students} = |A \cup B| + |A^c \cap B^c|$

Step-by-Step Calculation:

1. Calculate $|A \cup B|$: $|A \cup B| = 35 + 25 - 6 = 54$

2. Add the students who like neither: $\text{Total Students} = |A \cup B| + |A^c \cap B^c| = 54 + 3 = 57$

Final Answer:

The total number of students in the class is 57.
The correct option is A.

Would you like more details or help with similar problems?

Related Questions:

1. How would the calculation change if 10 students liked neither?
2. What if the problem included a third activity?
3. How is the inclusion-exclusion principle generalized for more than two sets?
4. Can you verify the total if given overlapping subsets in a Venn diagram?
5. What real-life scenarios can be modeled using the inclusion-exclusion principle?

Tip:

When solving problems involving sets, always check if there are overlaps (intersections) and exclusive groups (like "neither" cases) to ensure no double-counting!