In a class of47 student ,31 study physics ,26 study chemistry and 16 study Biology . 3study all the three subjects, 2 study chemistry and biology only and 6study Biology only
To find the number of students who study at least one subject, we need to add up the number of students who study each subject individually and then subtract the number of students who study all three subjects.
Number of students who study physics = 31
Number of students who study chemistry = 26
Number of students who study biology = 16
Number of students who study at least one subject = (31 + 26 + 16) - 3
= 73 - 3
= 70 students.
To find the number of students studying each subject, we can use the principle of inclusion-exclusion.
Let's define the following sets:
- P: the set of students studying physics.
- C: the set of students studying chemistry.
- B: the set of students studying biology.
We know that the total number of students is 47, and we need to find the number of students in each set.
First, let's start by finding the number of students who study both physics and chemistry, denoted by |P ∩ C|. We are given that 3 students study all three subjects.
|P ∩ C ∩ B| = 3 (since these students study all three subjects)
Next, we need to find the number of students who study both chemistry and biology, denoted by |C ∩ B|. We are given that 2 students study chemistry and biology only.
|C ∩ B| = |C ∩ B ∩ ¬P| + |C ∩ B ∩ P|
= 2 (since these students study chemistry and biology only, and we know they don't study physics)
Next, we need to find the number of students who study physics only, denoted by |P - C - B|. We can calculate this by subtracting the number of students who study physics and other subjects from the total number of students studying physics.
|P - C - B| = |P - C - B ∩ ¬(P ∪ C ∪ B)|
= |P - C - B| - |P ∩ C ∪ P ∩ B ∪ P ∩ C|
Next, we need to find the number of students who study chemistry only, denoted by |C - P - B|. We can calculate this by subtracting the number of students who study chemistry and other subjects from the total number of students studying chemistry.
|C - P - B| = |C - P - B ∩ ¬(P ∪ C ∪ B)|
= |C - P - B| - |P ∩ C ∪ P ∩ B ∪ P ∩ C|
Next, we need to find the number of students who study biology only, denoted by |B - P - C|. We can calculate this by subtracting the number of students who study biology and other subjects from the total number of students studying biology.
|B - P - C| = |B - P - C ∩ ¬(P ∪ C ∪ B)|
= |B - P - C| - |P ∩ C ∪ P ∩ B ∪ P ∩ C|
Finally, we can calculate the number of students studying each subject:
|P| = |P - C - B| + |P ∩ C ∪ P ∩ B ∪ P ∩ C| + |P ∩ C ∩ B|
|C| = |C - P - B| + |P ∩ C ∪ P ∩ B ∪ P ∩ C| + |C ∩ B ∩ ¬(P ∪ C ∪ B)|
|B| = |B - P - C| + |P ∩ C ∪ P ∩ B ∪ P ∩ C| + |C ∩ B ∩ ¬(P ∪ C ∪ B)|
Given that 31 students study physics, 26 students study chemistry, and 16 students study biology, we can substitute these values into the equations to find the missing values for |P - C - B|, |C - P - B|, and |B - P - C|.
After calculating the above equations, we can find:
|P| = 14
|C| = 10
|B| = 6