In a class of 45 students, 32 offer Physics and 28 offer Government. If 12 students offer neither subject, find how many offer both subjects. Draw a Venn diagram.

The number of students who offered at least one subject is $n(P \cup G)$.

$$n(P \cup G) = n(U) - n(P' \cap G')$$ $$n(P \cup G) = 45 - 12$$ $$n(P \cup G) = 33$$

We use the formula for the union of two sets:

$$n(P \cup G) = n(P) + n(G) - n(P \cap G)$$

Substitute the known values:

$$33 = 32 + 28 - n(P \cap G)$$ $$33 = 60 - n(P \cap G)$$ $$n(P \cap G) = 60 - 33$$ $$n(P \cap G) = 27$$

Number of students who offered Physics only:

$$n(P \cap G') = n(P) - n(P \cap G) = 32 - 27 = 5$$

Number of students who offered Government only:

$$n(P' \cap G) = n(G) - n(P \cap G) = 28 - 27 = 1$$

a) Draw the Venn diagram to represent the information.
A Venn diagram would show two overlapping circles, one for Physics (P) and one for Government (G), inside a rectangle representing the universal set (U).

• The intersection of P and G would contain 27 students.
• The part of circle P outside the intersection (Physics only) would contain 5 students.
• The part of circle G outside the intersection (Government only) would contain 1 student.
• The area outside both circles but inside the rectangle (neither subject) would contain 12 students. The sum of these parts is $27 + 5 + 1 + 12 = 45$, which is the total number of students.

b) How many students offered both subjects?
This is $n(P \cap G)$.

$$n(P \cap G) = \boxed{\text{27}}$$

c) What is $n(P \cup G)$?
This is the number of students who offered Physics or Government or both.

$$n(P \cup G) = \boxed{\text{33}}$$

d) What is $n(P' \cap G')$?
This is the number of students who offered neither Physics nor Government.

$$n(P' \cap G') = \boxed{\text{12}}$$

e) What is $n(P' \cap G)$?
This is the number of students who offered Government but not Physics (Government only).

$$n(P' \cap G) = \boxed{\text{1}}$$

f) What is $n(P \cap G')$?
This is the number of students who offered Physics but not Government (Physics only).

$$n(P \cap G') = \boxed{\text{5}}$$