A certain experimental mathematics program was tried out in 2 classes in each of 42 elementary schools and involved 55 teachers. Each of the classes had 1 teacher and each of the teachers taught at least 1, but not more than 3, of the classes. If the number of teachers who taught 3 classes is n, then the least and greatest possible values of n, respectively, are Choices
A) 0 and 13  B) 0 and 14  C) 1 and 10  D) 1 and 9  E) 2 and 8
One may notice that greatest possible values differ in each answer choice in contrast to the least values, which repeat.   To find out the greatest value you should count the total classes (42*2=84), then subtract the total #of teachers since we know from the question that each teacher taught at least one class (84-55=29). Thus we get a number of the available extra-classes for teachers, and all that we need is just to count how many teachers could take 2 more classes, which is 29/2 = 14.5. So the greatest possible value of the # of teachers who had 3 classes is 14.   Only answer 0 and 14 has this option.
<result>B</result>
A certain experimental mathematics program was tried out in 2 classes in each of 42 elementary schools and involved 55 teachers. Each of the classes had 1 teacher and each of the teachers taught at least 1, but not more than 3, of the classes. If the number of teachers who taught 3 classes is n, then the least and greatest possible values of n, respectively, are