The number of students that studied all three subjects is 32.
How to solve
Let's start by drawing a Venn diagram to illustrate the information given:
Let M represent students who studied Mathematics.
Let P represent students who studied Physics.
Let C represent students who studied Chemistry.
The total number of students (including those who studied none of the three subjects) is 120.
Given information:
Total students (M ∪ P ∪ C) = 120
Students who studied none of the three subjects = 22
Students studying Mathematics (M) = 60
Students studying Physics (P) = 40
Students studying Chemistry (C) = 55
Students studying Physics and Mathematics only = 12
Students studying Chemistry and Physics only = 8
Students studying Mathematics and Chemistry only = 7
From the Venn diagram, we can see that the intersection of all three subjects (M ∩ P ∩ C) represents the number of students that studied all three subjects.
To find the number of students that studied all three subjects:
Students in the intersection of all three subjects = Students in (M ∩ P ∩ C)
Using the Venn diagram, the number of students in (M ∩ P ∩ C) = 120 - (60 + 40 + 55 - 12 - 8 - 7 + 22)
Students in (M ∩ P ∩ C) = 120 - (120 - 32) = 32
Therefore, the number of students that studied all three subjects is 32.