AMM Problem 12546 Solution

In this article, I provide the solution to AMM Problem 12546. 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 2nd such article.

This problem has to do with an interesting coin flipping process. We start with $n$ coins with an extra coin $R$. The coins have the same bias, Heads with probability $p$ and Tails with $1-p$. We flip all of them simultaneously, but, whatever coins differ from $R$, we keep them aside and record their outcome. We continue till no coins remain. We have then recorded $n$ results. The question asks about the distribution of number of heads in the process.

I felt that this problem boils down very naturally to a recurrence after what happens on the first flip. Following this, I showed this is a forward recurrence.
I decided to convert this recurrence to a generating function instead of trying to directly solve it. The generating function satisfied a very unique looking functional equation which I had never solved or seen before.
Empirically, through simulations, I found a very interesting guess, which was against my intuition. But, plugging it into the generating function worked, and due to the recurrence being forward, this was indeed the unique solution.
This was my favorite problem that I tried in this set!

Click Here for the Solution

I hope you enjoyed reading the article

Any feedback is greatly appreciated!