Question 1191784: In a group of 200 students, each student studies at least one of the three science subjects:
Biology, Chemistry and Physics. 130 study Biology, 135 study Chemistry, 115 study Physics, 86
study Biology and Chemistry, 70 study Chemistry and Physics, and 64 study Physics and Biology.
Illustrate this information on a clearly labelled Venn diagram, showing the number of elements
in each separate region. Hence find the number of students who study
a.All 3 subjects
b. Exactly 2 subjects
c.only biology
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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
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