AMM Problem 12545 Solution
Analysis and Combinatorics: On the difference between two binomial terms
In this article, I provide the solution to AMM Problem 12545. This article is being
published a day after the AMM deadline!
This is part of my December 31st 2025 deadline, of which I solved 5/7 problems that
I'm sharing here! This is the 1st such article.
This problem has to do with comparing a difference of two terms:
\[ f_m(n) = \frac{\binom{n}{m}(n-1)^2 - (m+1)\binom{n}{m+2}}{(n-1)^{m+2}} \]
This question asks about the relation between $f_m(n)$ and $f_m(n+1)$, when one is larger than the other, and to show that they're never equal.Despite the presence of combinatorial terms, I believe that the theme of this problem is related to analysis instead.
It ended up being quite a long bash, since I decided to reduce everything to polynomials and comparing gradients using mathematical software to verify my calculations (Wolfram Alpha).
I think it's likely that a clean combinatorics/probability based solution exists, but given the answer of this problem, it seems unlikely at least to me.
It's still quite an enjoyable read!
I hope you enjoyed reading the article
Any feedback is greatly appreciated!