The image presents a set theory problem involving 130 students studying various subjects: mathematics, physics, chemistry, and biology. Let's break down the problem and solve each part step by step.

Problem Breakdown:

• Total number of students = 130
• 60 students study mathematics.
• 51 students study physics.
• 30 students study both mathematics and physics.
• 54 students study chemistry.
• 26 students study both mathematics and chemistry.
• 21 students study both physics and chemistry.
• 12 students study all three subjects: mathematics, physics, and chemistry.
• All students who study neither mathematics nor physics are studying biology.

Questions:

1. How many are studying biology?
2. How many not studying chemistry are studying mathematics but not physics?
3. How many students are studying neither mathematics, physics, nor chemistry?

1. How many are studying biology?

To find the number of students studying biology, we first need to determine how many students are studying mathematics or physics. We will use the principle of inclusion and exclusion.

• Students studying only mathematics = $60 - 30 = 30$
• Students studying only physics = $51 - 30 = 21$
• Students studying both mathematics and physics = 30

Thus, the total number of students studying mathematics or physics is:

$30 \ (\text{only math}) + 21 \ (\text{only physics}) + 30 \ (\text{math and physics}) = 81$

Therefore, the number of students who are studying biology (i.e., neither mathematics nor physics) is:

$130 - 81 = 49$

So, 49 students are studying biology.

2. How many not studying chemistry are studying mathematics but not physics?

First, let's calculate how many students are studying mathematics but not physics or chemistry.

• Students studying only mathematics = $60 - 30 - 26 + 12 = 16$

So, 16 students are studying only mathematics and not physics or chemistry.

3. How many students are studying neither mathematics, physics, nor chemistry?

The number of students studying mathematics, physics, or chemistry can be calculated using the inclusion-exclusion principle.

Define:

• $M$ = 60 (students studying mathematics)
• $P$ = 51 (students studying physics)
• $C$ = 54 (students studying chemistry)
• $M \cap P = 30$
• $M \cap C = 26$
• $P \cap C = 21$
• $M \cap P \cap C = 12$

Using inclusion-exclusion:

$\text{Total in M, P, or C} = M + P + C - (M \cap P) - (M \cap C) - (P \cap C) + (M \cap P \cap C)$ $= 60 + 51 + 54 - 30 - 26 - 21 + 12 = 100$

Thus, 100 students study mathematics, physics, or chemistry. Therefore, the number of students who study none of these subjects is:

$130 - 100 = 30$

So, 30 students are studying neither mathematics, physics, nor chemistry.

Summary:

1. 49 students are studying biology.
2. 16 students not studying chemistry are studying mathematics but not physics.
3. 30 students are studying neither mathematics, physics, nor chemistry.

Would you like a detailed breakdown of any specific part? Here are five questions you might find interesting:

1. How does the inclusion-exclusion principle apply in more complex set theory problems?
2. Can we use Venn diagrams to visualize the relationships in this problem?
3. What if the total number of students were different, how would that change the calculations?
4. How would the solution change if the overlap between chemistry and physics were larger?
5. Could we apply the same methods to different subjects or scenarios?

Tip: When dealing with multiple overlapping groups, the inclusion-exclusion principle helps avoid overcounting when combining sets!