HELP

[AJWILL]

If 37 teachers are to be assigned to 64 classes in such a way that each of teacher teaches at least one class and at most three classes. what are the greatest possible number and the least possible number of the teachers who teach three classes?

[kartikshah]

Situation 01: Greatest possible number of teachers who teach three classes
Let all 37 teachers first be assigned 1 lecture each.
Remaining lectures = 64 - 37 = 27 lectures
In order to assign two more lectures to each teacher, divide the remaining lectures in pairs of 2 = 27/2 = 13 full pairs + 1
Therefore, 13 teachers will get 3 lectures, 1 teacher will get 2 lectures and the remaining 23 teachers will get 1 lecture each.


Situation 02: Least possible number of teachers who teach three classes
Let all 37 teachers first be assigned 1 lecture each.
Remaining lectures = 64 - 37 = 27 lectures
Assign one more lecture to each teacher. Now remaining lectures = 0
Therefore, 27 teachers will get 2 lectures and 10 teachers will get 1 lecture each while NO teacher will get 3 lectures.