Question

There are 19 people in a class, and Jonathan solved 9 problems more than anyone else. Can we say for certain that there must be at least 3 students in the class who solved the same number of problems as each other?

A. Yes
B. No

Answer 1

Yes, there must be at least 3 students in the class who solved the same number of problems as each other.

Step-by-step explanation:

Yes, we can say for certain that there must be at least 3 students in the class who solved the same number of problems as each other.

Since there are 19 people in the class, and Jonathan solved 9 problems more than anyone else, we can assume that Jonathan solved the most problems. Let's assume Jonathan solved x problems. Then the number of problems solved by the other 18 students will be at most x-9, since Jonathan solved 9 more problems than anyone else.

For the other students, there are only x-9 possible different numbers of problems they can solve. However, since there are 18 other students, and only x-9 possible different numbers, by the Pigeonhole Principle, there must be at least two students who solved the same number of problems. And since Jonathan already has the highest number of problems, there must be at least 3 students who solved the same number of problems as each other.