Pigeonhole principle: prove that a class of 21 has at least 11 male or 11 female students.

Question

Here is the problem in full with no other special restrictions:

"If there are 21 students in a class, show that at least 11 must be male or female."

Answer 1

Let us proof by contradiction. By assuming that there are only 10 boys and 10 girls, we attempt to not fulfill the requirement. However, there are 21 people, so there must be one more boy or girl, meaning that there would be 11 boys or 11 girls.

To answer the comment, we also proof by contradiction. Let us say there are less than 5 males. That would mean that there will be more than 21-5=16 females. Hence it is proven that there will be at least 5 males or 16 females.

Answer 2

Assume there are at most 10 male students and at most 10 female students. Then there are at most 20 students, a contradiction.