November 21, 2008 | 25 CommentsThe stock market is crashing and may have even bottomed out. Naturally, people are beginning to look around for buying opportunities. I have the perfect one.

Hearing aids.

Any company that sells or markets hearing aids is positioned for rapid growth over the next ten to thirty years. My scientific estimate is that a dollar invested today will bring twenty a few years from now. I give you this hot tip for free, my friends, just for being a loyal reader of this blog. You simply cannot go wrong.

My evidence for this stunning opportunity is based on my experiences commuting via the F train. My insight arrived suddenly, when yesterday I was surrounded by at least seven people wearing Thinking Suppression Devices (TSDs). I had to give up on reading James Fitzjames Stephen because each commuter’s device was louder than the others’. Everybody was wearing earphones, but even still, I was assailed by various booms and crashes which were clearly audible to anybody—to anybody not wearing a TSD themselves. These people are obviously going to lose a significant part of their hearing, if not go deaf, hence my advice about buying stock in hearing aid companies.

But I decided to make this an opportunity instead of despairing and following my usual approach of looking at my fellow passengers like they had just stepped in what they were listening to.

Now, the only distinct sounds I could make out—and I listened attentively—were various combinations of *boom*s and *chhh*s (electronic cymbal crashes). This was my first important clue.

I asked the guy sitting next to me, who was fiddling with his TSD, “How many songs does that thing hold?” He said a number in the thousands, and that he had just about filled it up and was looking into getting a better model. I asked, “Got any Duke Ellington on it?” When he said “Who?”, I had my second clue, and realized that my *boom*—*chhh* observation was not an accident. I therefore told him that he didn’t need a larger device, because there were only about 100 pop songs in the world anyway.

“Yes,” I sagaciously intoned, “based on my scientific measurements of your music—this is only an estimate, to be confirmed later, mind you—there are only about 100 possible songs that can be made. You don’t need a more expensive device. Think of the money I have just saved you!”

The look he gave me told me he was deeply aware of my genius. The fact that he edged toward the door as fast as he could, so that he could jump out at the next stop told me how excited he was to pass on his new knowledge to his friends and family.

It was on that train ride that I devised the Combinatorial Theory of Finite Musical Variety. **This theory states that there are exactly, and only, 124 possible pop songs.** Here is how it is proved.

Our first piece of evidence is that every popular song is comprised *solely* of combinations of *boom*s and *chhh*s. There are other trivial elements, related to the outmoded and ancient theories of melody, harmony, point and counterpoint, and lyricism, but these can be, like they now always are, ignored.

The second piece of evidence is that, to build a pop song, you must have either three, four, or five *boom*s and *chhh*s in combination and that at least one of these must be a special *boom*. Examples:

*boom* *boom* *chhh*
*boom* *boom* *boom* *chhh*
*boom* *chhh* *chhh* *boom* *chhh*

Ready? Let’s start counting. Begin with the three combination, with one mandatory *boom*. Suppose first that the *boom* is at the end. The other two slots must be filled with *boom*s and *chhh*s. We could have two *boom*s, two *chhh*s, or one of each with either leading the way. That’s 4 possibilities. But then we remember that we have the mandatory *boom* at the end, which could be placed in any of the three available slots (it could have come first, second, or third), and then the other *boom*s and *chhh*s would fill in the other two slots. This makes 3 x 4 = 12 possibilities.

What if there were 4 slots and one mandatory *boom*? Suppose again that the regulated *boom* is at the end, and the *boom*s and *chhh*s fill in the other three slots. They could all be *boom*s or all *chhh*s, or various combinations. We could list them all, but that gets to be a pain, so let’s use some math. We have three slots and we could put 0, 1, 2, or 3 *boom*s in those slots, with *chhh*s filling in the blanks.

There is a formula for this called *choose*. If we say “3 choose 0”, we say we have three slots and we choose no *boom*s to go in them. The *choose* formula tells you the number of ways we can do this. It should be obvious to you that the only way to place no *boom*s out of three is 1; namely, you choose no *boom*s. Formally, the answer is

n!

—————–

(n-k)! k!

where n! is read “n factorial” and means “n x n-1 x n-2 x … x 2 x 1”. In our example, n = 3 for three slots. At first, k = 0 for no *boom*s. The formula works out to be

3 x 2 x 1

———————–

(3 x 2 x 1) x 0!

where you have to know that “0! = 1” (there are good, technical, mathematical reasons for this, which I’ll skip). There answer equals 1. Then you work this formula out for k = 1, k = 2, and k = 3 because we want to know how many different *boom*s go into the 3 slots for each possible number of *boom*s.

The answer turns out to be 1 + 3 + 3 + 1 = 8 for k = 0, k = 1, k = 2, and k = 3. But again we have to remember that one mandatory *boom* could have gone into any of the four slots, so we have to multiply the answer by 4 to get 32.

Thus far, we have 12 + 32 = 44 possible songs, but we still have to calculate the number of possibilities if there were 5 slots and one mandatory *boom*. The math is exactly the same, except n = 4, and k = 0, k = 1, k = 2, k = 3, and k = 4. But it turns out that there is a simplification when you sum up all possible combinations of 2 things (*boom*s and *chhh*s). The sum of “n choose 0” + “n choose 1” + … + “n choose n” = 2^{n}, which is very easy to calculate.

If n = 4 then 2^{4} works out to 16 different songs with the mandatory *boom* at the end, but there are 5 possible places for that *boom*, so we have 5 x 16 = 80 combinations.

In total, then, we have 12 + 32 + 80 = 124.

That’s it, friends! There are only 124 possible pop songs. I was therefore right to harangue my train-mate with this wisdom so that he would not waste his money on a device that holds more than this. It is therefore a matter of great curiosity how people could not have recognized this before now, a fact which can only tell us of the deleterious effect of pop music on the brain.

Incidentally, a corollary to the theory is: Each pop song must be played as loudly as possible, not only to annoy those around you, but to destroy your hearing as quickly as possible.

Incidentally, incidentally, I cannot vouch that the math here is 100% accurate. This is because, as I write this, the *beaujolais nouveau* arrived in stores this afternoon, and I am liberally sampling this year’s offerings.