Answer to Question #271508 in Discrete Mathematics for Aarush

Question

State the Pigeonhole Principle. In a result sheet of a list of 60 students, each marked “Pass” or “Fail

“. There are 35 students pass. Show that there are at least two students pass in the list exactly nine

students apart. (for example students at numbered 2 and 11 or at numbered 50 and 59 satisfy the

condition).

Answer 1

pigeonhole principle states that if n items are put into m containers, with n>m, then at least one container must contain more than one item.

There are 25 students "fail". There are 30 pairs of students in the list exactly nine

students apart. So, by pigeonhole principle, there are at least 2 pairs of students "pass" in this list.