## A random math illiteracy moment August 9, 2009

Posted by lumixedia in combinatorics, General.
Tags: , ,

I can’t figure out how, but this memory from policy debate camp a couple years back just floated into my head and this seems like the appropriate place to recount it.

This was back when the topic for the upcoming year was Resolved: The United States federal government should substantially increase its public health assistance to Sub-Saharan Africa. For readers not familiar with policy debate, policy teams generally come up with a single plan which falls under the scope of the resolution and research it in great depth throughout the year. A plan for this resolution might look something like “The United States federal government should substantially increase funding for treatment of AIDS in Zimbabwe”. In each round, the affirmative team presents their plan and the negative team attacks it.

Besides directly arguing that the plan is a bad idea, the negative team can also argue that the plan doesn’t actually fit the resolution by some interpretation of the resolution. This is known as a topicality argument and might go something like “Your plan to treat AIDS in Zimbabwe is non-topical because the resolution implies that the increased public health assistance should go to all of Africa, not just a small subset such as a single country”. This particular example is the subsets topicality argument.

Not surprisingly, in order for the negative team to win a topicality argument, they should demonstrate that their interpretation of the resolution should be preferred over other interpretations. One way to do this is to complain that a broader interpretation which includes the affirmative team’s plan would be unfair because it would make the research burden on the negative team impossible. For example, “Let’s say the affirmative team could choose to aid any subset of the countries in Sub-Saharan Africa. There are 48 countries in Sub-Saharan Africa. Then for each basic plan like ‘treat AIDS’, we would have to research 48 factorial plans corresponding to each possible subset the affirmative team might choose. 48 factorial is ridiculously big. Judge, vote negative.”

And since you’re the kind of person who reads this blog, you probably just blinked and said, “Wait”.

So the point of all this buildup was that back at the camp I went to, one of our instructors got the impression that 48! was the number of subsets of Sub-Saharan countries. This number proceeded to spread throughout camp, tossed off casually whenever the subsets topicality argument came up in conversation or lecture. I actually succeeded in explaining why the real number was 2^48 (or 2^48-1 I guess) to a couple fellow students once, but I doubt anyone cared enough to correct it in future arguments.

Lucky for debaters who like subsets topicality that 2^48-1 is still ginormous enough for the argument to stand (assuming everything else about it is logical, which is of course debatable). But still it’s really quite a lot smaller than 48!, which would be the number of plans involving sending assistance to each Sub-Saharan country one at a time. The number of plans that send assistance to any subset of the Sub-Saharan countries in a particular order would, I guess, be $\sum_{i=0}^{47}\frac{48!}{i!}$. I don’t see a closed formula for this at first glance. Darn.

I’m talking about really smart people here. This isn’t like those “punching 2×3 into the calculator” stories. This was a pretty intensely academic camp…it just happened to be a debate camp rather than a math/science camp. This is how acceptable it is to say silly things about math the moment you’re not in an extreme math/science environment.

Moral of the post: don’t toss around big numbers like they all belong to some vague big number cloud and that’s the only important thing about them. It makes people who like combinatorics sad.

For computing the number of ways to order a subset you don’t want a closed form; the more useful thing to have is the exponential generating function, which is $\frac{1}{1 - x} e^x$. This lets you access the asymptotic information.