To find the number of students who took none of the three subjects, we can use the principle of inclusion-exclusion.
First, let's add up the number of students who took each subject:
PE: 28 students
BIO: 31 students
ENG: 42 students
Next, let's subtract the number of students who took two subjects:
PE and BIO: 9 students
PE and ENG: 10 students
BIO and ENG: 6 students
Now, let's subtract the number of students who took all three subjects:
PE, BIO, and ENG: 4 students
To find the number of students who took none of the three subjects, we subtract the total number of students who took any of the subjects from the total number of students:
Total students - (PE + BIO + ENG - PE and BIO - PE and ENG - BIO and ENG + PE, BIO, and ENG)
100 - (28 + 31 + 42 - 9 - 10 - 6 + 4) = 100 - 90 = 10
Therefore, 10 students took none of the three subjects.