Question 1191784
Here's how to solve this problem using a Venn diagram:

**1. Set up the Venn Diagram:**

Draw three overlapping circles representing Biology (B), Chemistry (C), and Physics (P).

**2. Use the Principle of Inclusion-Exclusion:**

Let:
* |B| = Number of students studying Biology = 130
* |C| = Number of students studying Chemistry = 135
* |P| = Number of students studying Physics = 115
* |B ∩ C| = Number of students studying Biology and Chemistry = 86
* |C ∩ P| = Number of students studying Chemistry and Physics = 70
* |P ∩ B| = Number of students studying Physics and Biology = 64
* |B ∩ C ∩ P| = Number of students studying all three subjects (what we need to find)

The Principle of Inclusion-Exclusion for three sets is:

|B ∪ C ∪ P| = |B| + |C| + |P| - |B ∩ C| - |C ∩ P| - |P ∩ B| + |B ∩ C ∩ P|

We know that all 200 students study at least one subject, so |B ∪ C ∪ P| = 200. Plugging in the values:

200 = 130 + 135 + 115 - 86 - 70 - 64 + |B ∩ C ∩ P|
200 = 160 + |B ∩ C ∩ P|
|B ∩ C ∩ P| = 40

**3. Fill in the Venn Diagram:**

* **B ∩ C ∩ P:** 40 (all three subjects)
* **B ∩ C only:** 86 - 40 = 46
* **C ∩ P only:** 70 - 40 = 30
* **P ∩ B only:** 64 - 40 = 24
* **B only:** 130 - 46 - 40 - 24 = 20
* **C only:** 135 - 46 - 40 - 30 = 19
* **P only:** 115 - 24 - 40 - 30 = 21

**4. Answers:**

* **a. All 3 subjects:** 40 students
* **b. Exactly 2 subjects:** 46 + 30 + 24 = 100 students
* **c. Only Biology:** 20 students