In my Combinatorics class yesterday, my professor defined The Pigeonhole Principle (TPP) as follows:
"If you have $n$ holes into which you wish to put more than $mn$ pigeons, then some hole must contain more than $m$ pigeons."
I don't know how to interpret this definition. I've seen TPP before in a Discrete Math class I took a while ago, but it was defined differently and I'm pretty sure I remember something about 2$\lfloor n/k \rfloor$ + 1 being relevant.
Why are $m$ and $n$ being multiplied? Why do you need more than $m$ pigeons? Why not more than $mn$ pigeons?
The Wikipedia article on TPP has a nice example: https://en.wikipedia.org/wiki/Pigeonhole_principle
In it they have $m$ = 10 pigeons and have $n$ = 9 pigeonholes. However, according to the definition provided by my professor, I am wanting to put more than 90 pigeons into the 9 pigeonholes, so that means at least one hole must contain more than 10 pigeons. I don't see 90+ pigeons in Wikipedia's example.
If somebody could explain all this for me and why $m$ and $n$ are being multiplied that would be seriously amazing. Thank you so, so much.