The Renewal Equation

Yesterday I watched another lecture from The Joy of Math, this one about combinatorics. One of the questions the lecturer brought up was how many of the possible five-card poker hands contain at least one ace. I paused the lecture at that point to outline how I would approach that problem. I’m pleased to say that I didn’t come up with the incorrect approach that he went on to warn his audience against. On the other hand, I didn’t come up with the clever one he ended with, either.

I want to play with the question a bit more. My method was to find the number of possible hands containing one ace, the number of possible hands containing two aces, the number containing three aces, and the number containing four aces, then add those together. This gives a correct solution, but the clever approach is instead to find the number of possible hands that don’t contain an ace and subtract it from the total number of possible hands. Clearly these methods should give the same answer, and they do, but it’s not obvious to me that the two expressions involved are equal, and I want to see if I can gain any insights about why they are.

That is, why are these things equal:¹

$${4 \choose 1}{48 \choose 4}+{4 \choose 2}{48 \choose 3}+{4 \choose 3}{48 \choose 2}+{4 \choose 4}{48 \choose 1}$$

$${52 \choose 5}-{48 \choose 5}$$

(The expression ${52 \choose 5}$ is read “fifty-two choose five” and represents the number of ways to select 5 from among 52 possibilities, ignoring order.)

This is one of several avenues of exploration I’ve thought of while watching The Joy of Math, and thinking of it yesterday brought me back to another question I’ve been puzzling over. That is, what is the best way to keep track of such things?

Right now, my math work is in four composition books: one book for problem sets from my review of calculus and its prerequisites, one quad-ruled book for graphs associated with those problem sets, one book for problem sets from proofs-based math textbooks, and one book for exploration and puzzles. I don’t have a settled place to record things to explore in the future, however. In 2024, I was using this blog for that, tagging each post that contained an unresolved question with the “to-do” tag. That doesn’t work as well when I’m not blogging every day, though, and it was a bit unwieldy already. I’ve thought about starting yet another book. I have a small journal that would serve the purpose. That seems a bit like overkill, though.

Perhaps I should move exploration and puzzles onto loose sheets in a three-ring binder? Then I could write each new topic on a new sheet and add sheets behind it whenever I needed more room to work on the problem. That would also allow me to dispense with the acceptable but not ideal system of page number references I’m currently using to organize the exploration and puzzle book.

On the other hand, I don’t really like three-ring binders. It’s too easy for them to lose pages. And I do like composition books, both because they don’t have that issue and simply for their aesthetic appeal as objects.

Anyone have any other ideas?

1. To do. ↩︎