2nd January 2026

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:

fm(n)=(nm)(n−1)2−(m+1)(nm+2)(n−1)m+2 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 fm(n)f_m(n) and fm(n+1)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!

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