Out of 30 students in college 15 takes art courses, 8 takes biology courses and 6 takes chemistry. it is also known that 3 students take all the 3 courses. show that 7 or more students taken none of the course?

15, 8 and 6. all but one subtract 3 from, representing the 3 students who take all three. don't subtract the 3 from one of them, they're still in the class.

so 15, 5 and 3. 15 + 5 + 3. 23. 23 students take the 3 courses, but 7 people take none of them.