Unequal exam
120 students sit an exam with 100 marks available. What is the smallest number of pairs that are awarded the same mark?
Age
11 to 14
Challenge level
Problem
One hundred and twenty students take an exam which is marked out of $100$ (with no fractional marks). No three students are awarded the same mark.
What is the smallest possible number of pairs of students who are awarded the same mark?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
$19$ is the smallest possible number of pairs of students who must be awarded the same mark.
This occurs when every possible mark from $0$ to $100$ is awarded to at least one student. This accounts for $101$ students and therefore the remaining $19$ students must all be awarded the same mark as exactly one of their colleagues.
The $120$ students are made up of $19$ pairs of students who are awarded the same mark and $82$ students who are all awarded a different mark from everyone else.