At most how many students are there in the group? There is a group of students. Each of them has to join at least one of the football class, piano class, French class and art class. The maximum number of students for each class is 9. if every two students would meet each other in at least one class, at most how many students are there in the group?
 A: The maximum would be 15. To see why, create 4 boxes to represent the classes (football, french, art, piano), each of which has a maximum capacity of 9 students. 
Fill one of the boxes with the maximum number of students - that is 9 students labeled A through I, that is: {A,B,C,D,E,F,G,H,I}. 
Now you have 3 classes remaining to fill, and you want to find the maximum number of new elements you can add, while ensuring that all of A-I are still in a class with each of new person added. So partition the first class evenly among all three of the remaining classes, placing {A,B,C} in one class, {D,E,F} in another, and {G,H,I} in the last one. You can then fit new students {J,K,L,M,N,O} into each of these classes, ensuring that they all share a class with each other and every other student from the first class.
Now suppose you want to add another student P. All of the classes are full, and you'll find that there is no student you could remove from a class to make space for them, without making it so that there is at least one pair of students that don't share any classes.
