 If I had to choose between some bird waking me up at 6am every morning or being deathly ill for two days, I'd probably take one flu over the cuckoo's nest. Imagine you're sitting at a round table at some party with your beverage of choice and someone makes a toast. When the toast is finished, you're expected to conclude with a cheers, clinking your glass against everyone else's at the table. With two or three other people, no big deal, but six or seven, your throat starts to get kind of parched by the time you're finished with all the clinking. Also, although the number of clinks you have to get through in order to drink your beverage only increases by one every time another person is added to the table, the total number of clinks grows by one for every person already at the table. Three people, three clinks. Four people, six clinks. Five people, ten clinks. If you have any familiarity with mathematics, your first suspicion might be that this growth pattern is exponential because it accelerates the bigger it gets. But we've actually left exponentials in the dust. We're in the realm of combinatorial explosion, where the factorial operator reigns supreme. Compared to the rocket ship that is n-factorial, the normally mind-blowing growth rate of a simple exponential function looks like a doddering snail who isn't in a particular hurry. Combinatorial explosion, like that of clinks during cheers, is the basis of the birthday paradox, a favorite example of mathematicians specializing in combinatorics. How many people do you need in a group before you can be more than 50% sure that two of them share a birthday? Most people think the number must be something like 180, but they don't realize that birthdays work like clinks. Every person that you add to the group doesn't just add one more chance for a shared birthday. It adds one more chance per person already in the group. Once you get to just 23 people, odds are that two of them were probably born on the same day of the year. That might seem unintuitively low, but if we were all sitting at a very large table together, we'd have to clink our glasses 185 times in total, more than half the number of days in a year. This sort of combinatorial explosion shows up in all sorts of places, from computer science to warfare. But I want to focus on one example that's particularly relevant right now, at the beginning of cold and flu season. Epidemiology and herd immunity. Influenza is an adaptive little family of critters that we've only recently developed any capacity to fight with medicine. It mutates in ways that are hard to predict and which always seem to find purchase in human hosts. You're probably fortunate enough to live in an area where a flu is usually an inconvenience, maybe making you feel like garbage for a few days if you're unlucky enough to catch it, but so long as you have a healthy immune system, nothing serious. But that's not always the case. Last year, 900,000 people were hospitalized for flu-related symptoms and complications. 80,000 people in the United States died, including hundreds of pediatric deaths, as children under the age of five are at significantly increased risk. A bad flu can easily find purchase in people with compromised immune systems and breed until it overwhelms the remaining defenses, rendering them vulnerable to the fatal hemorrhagic pneumonia that stops them from breathing. Those figures are partially why any doctor worth their salt will get on your case to get a flu shot and to stay home if you come down with a full-blown illness. Most of us treat flusies in with a sort of cavalier acceptance because it's usually not a huge deal even if we do catch it, but doctors are aware that we're locked in a perpetual war against this organism and that war has fatalities. The flu shot has a ton of clear medical benefits and almost no drawbacks. It obviously decreases the chances that you'll contract the flu if you're exposed to it by an inconsiderate coworker or a checkout clerk who can't afford to take a day off. If you do catch it, it shortens the average length of time that you'll feel like crap, which is nice. Many people imagine that they've gotten the flu from a shot, but the substances used in flu vaccines are very tightly controlled, are injected into a location that isn't really conducive to the flu's replication and really can't give you the flu. It just turns out that around the time that people get their shots, they also happen to be exposed to a lot of active flu virus. But the real reason doctors get up in arms about flu shots doesn't really have to do with whether or not you specifically get sick. It has to do with that principle of combinatorial growth. Let's start down at the flu's eye view at the cellular level. Viruses aren't really intelligent in any sense of the word. They're essentially just conveniently shaped packages of protein with a DNA blueprint inside for making more, which means that they're not great at doing their thing. They don't have any sophisticated abilities for movement or targeting. Their mechanism of infection is stochastic, depending solely on the law of large numbers and things happening to bump into each other in precisely the right way at random. Every individual virus has a snowball's chance in hell of delivering its payload. But with enough snowballs, maybe one manages it. Of course, that singular event isn't enough to get you sick. Even a full cells worth of flu virus exploding out into your body faces the same astronomically slim odds of everything going exactly right to replicate again. However, that probability shoots up with every subsequent cell that gets infected because of combinatorial explosion. Just like clinks of glasses, each additional lucky virus doesn't just add one more chance for the virus to get a foothold, but a whole set of interactions with every other cell that that first cell comes in contact with. The flu shot can't guarantee that that runaway process won't happen, but it can increase the inertia that it takes to get the ball rolling and slow its acceleration. Even if a group of cells gets unlucky and starts manufacturing flu, the antibodies created by the flu vaccine act on the same principle. They increase the number of stochastic interactions that will be fatal to the virus. This is why flu duration is decreased if you get your shot, even if you do get sick. Now, if we zoom out, this picture is surprisingly accurate for the spread of communicable disease through a society, replacing the cells with individual humans. Every interaction that you have with someone carrying the virus, whether they're visibly sick or not, is a role of the dice. If only a couple people are carrying the disease, it's actually pretty unlikely that you'll run into one in the first place, let alone get sick. But as the number of carriers climbs and the number of carriers you encounter, the number of chances that you have to be unlucky enough to contract the virus gets higher and higher. The flu shot has a similar retarding effect at this scale via something called herd immunity. The idea is to limit the spread of the disease by retarding its opportunities for infection. Rather than depending on each individual person's immune system to repel the flu, vaccinated people act as a buffer between the infected and more potential targets for infection. Someone who's vaccinated isn't invulnerable, but they're not participating fully in the combinatorial explosion. They're not really clinking with everyone else and don't carry infection as readily from person to person as a result. The effects of herd immunity are dramatic. In 1995, the U.S. began a program of chickenpox vaccination for children over the age of five, the youngest age that had been determined to be safe from complications due to the vaccine. Obviously, those children showed a decrease incidence of chickenpox, but the incidence of the disease for children under five who weren't vaccinated fell almost 90% by 2008. There were still children over five who weren't vaccinated and there were still kids who got chickenpox. The only thing that changed was that numerous children who might have otherwise gotten chickenpox were removed from the equation and didn't add their whole network of interactions to the total number of opportunities for the disease to spread. As a result, the number of infant mortalities due to chickenpox went from about 50 per year to zero. Look, I get it. You're a busy person. You almost never get sick, just a few sniffles at worst and that doesn't seem to depend on whether or not you get the flu shot. You've probably never known anyone who's been hospitalized for flu-related complications and nobody with a compromised immune system has given you grief for not getting immunized. But even if you feel like your amazing constitution is sufficient protection for you, we are fighting a desperate numbers game against the power of combinatorial explosion and every tiny advantage that we can muster has massive consequences. If you can avoid being communicable, even for the half day it takes for your awesome immune system to eradicate the last vestiges of flu exposure, that's a couple fewer clinks and that can really make a difference. Please, get your flu shot. What instances can you think of where combinations of individual entities are what result in enormous growth? Please leave a comment below and let me know what you think. Thank you very much for watching. Don't forget to blah, blah, subscribe, blah, share and don't stop thunking. Also, get your flu shot. Just go. I mean, it takes like five minutes. It's free at most pharmacies. Just get a flu shot. Do it.