Question

In a class of 55 students 21 students studies physics 24 studies geography and 23 study economics if six students study both physics and geography and eight study both geography and economics and five studies both economics and physics if x studies all three subjects and two eggs studies none of the three subjects find the value of x

Answer 1

To find the value of \( x \) (the number of students studying all three subjects), we can use the principle of inclusion-exclusion.

Let:
- \( P \) = number of students studying Physics = 21
- \( G \) = number of students studying Geography = 24
- \( E \) = number of students studying Economics = 23
- \( P \cap G \) = number of students studying both Physics and Geography = 6
- \( G \cap E \) = number of students studying both Geography and Economics = 8
- \( E \cap P \) = number of students studying both Economics and Physics = 5
- \( x \) = number of students studying all three subjects (Physics, Geography, and Economics)
- \( N \) = total number of students = 55
- \( N_{\text{none}} \) = number of students studying none of the subjects = 2

First, we will calculate the number of students studying at least one subject:
\[
N_{\text{at least one}} = N - N_{\text{none}} = 55 - 2 = 53
\]

Now, using the principle of inclusion-exclusion, the number of students studying at least one subject can be expressed as:
\[
N_{\text{at least one}} = P + G + E - (P \cap G) - (G \cap E) - (E \cap P) + (P \cap G \cap E)
\]
Substituting the known values:
\[
53 = 21 + 24 + 23 - 6 - 8 - 5 + x
\]
Simplifying this equation:
\[
53 = 21 + 24 + 23 - 6 - 8 - 5 + x
\]
\[
53 = 68 - 19 + x
\]
\[
53 = 49 + x
\]
Now, solving for \( x \):
\[
x = 53 - 49
\]
\[
x = 4
\]

Thus, the value of \( x \) (the number of students studying all three subjects) is \( \boxed{4} \).