Let's assume the minimum number of students needed to take the course is x.
We know that the total cost for the course is $275.
Since the instructor is also included in the maximum number of people the room can hold, we have to subtract 1 from the maximum capacity to get the number of students.
So, the number of students in the room will be x - 1.
To find the cost per student, we divide the total cost by the number of students: 275 / (x-1).
We need this cost to be less than $15, so we can set up the inequality:
275 / (x-1) < 15.
To solve this inequality, we start by multiplying both sides by (x-1) to get rid of the denominator: 275 < 15(x-1).
Simplifying the right side: 275 < 15x - 15.
Adding 15 to both sides: 290 < 15x.
Dividing both sides by 15: 19 < x.
So, the minimum number of students needed to take the course is 19. Answer: \boxed{19}.