How do I use the Pigeonhole principle to show that in a class of 25 students where every student is either a sophomore, freshman or a junior there are at least 9 sophomores or 9 seniors or 9 juniors ?
My solution:
Assume by contradiction that
there are 8 sophomores, 8 freshman and 8 juniors
then
$$ 8+8+8 = 24 \neq 25 $$
thus by contradiction there are at least 9 sophomores or 9 seniors or 9 juniors
Is this solution a correct application of the Pigeonhole principle ?
Thank you in advance guys!
Best Answer
This is pretty much correct.
The one nitpick I have is that, if there are not $9$ of a given year, there are at most $8$ of the given year. That is, $8$ is simply the worst case scenario.