Mathematics and chemistry are both subjects that people shy away from, due to their complexity and abstract nature. The purpose of this writeup, in a way, is to drive right through that fear and end up on the other side. I will be explaining a few concepts of organic chemistry in such a way to demonstrate that the field is too vast to ever be understood.

In Mathematics, "Factorial" is the term for the number of possible outcomes of random, exclusive events. Or put more simply, factorial equations tell you how many ways you can draw different colored marbles out of a bag. If you have a bag with a red, green, yellow and blue marble, you can draw any of the four out at first, followed by any of the remaining three, either of the remaining two, and finally the last marble. Mathematically, this gives us 4*3*2*1 combinations, which is 24 combinations. One of the most noticeable things about factorial combinations is how quickly they pile up: while there are 24 combinations of 4 colors of marble, there are 6*5*4*3*2*1 or 720 combinations of 6 color, and 40320 combinations of 8 colors. Factorials have a wide application in both pure mathematics, and can be applied to a number of different problems.

Organic chemistry is the study of carbon based compounds. Carbon has the ability to form complex structures of rings and chains, and organic chemistry is the base of the chemistry of life, as well as of many other things. Because carbon chains can be extended and bent and manipulated in so many ways, they can be endlessly added to, like LEGO Bricks. Non-carbon atoms can be added to a basic carbon structure, and these are called radicals. The different combinations with which radicals can be added to a carbon structure is responsible for the endless diversity of organic chemistry.

Lets take a look at a basic organic molecule, in this case, benzene, 6 carbon atoms in a ring with each one bonded to a hydrogen atom.

```
_
/ \
\_/

```
Pretty simple, right? Now lets replace one of those hydrogen atoms (which are by convention left unwritten, because the carbon-hydrogen bond is so ubiquitous) with a halogen atom, in this case fluorine.
```
_
/ \
\_/\F

```
Since the molecule can rotate freely, there is only one possible type of Fluorobenzene. But lets add another fluorine atom.
```
__F
/ \
\_/\F

F__
/ \
\_/\F

F  _
\/ \
\_/\F

```
Here we have three different forms of the same isomer: these are called the ortho, meta and para forms of the compound. Since the molecule can rotate freely, the other two positions would be identical to the first and second forms. So far we have three possibilities for difluorobenzene. Three is not an overwhelming number, so lets add a twist: another halogen atom.
```

__Br
/ \
\_/\F

```
By adding an atom of Bromine, we now have three additional possibilities. And lets add another atom, an atom of Chlorine:
```

Cl___Br
/ \
\_/\F

```
Now that there are three different atoms, the atom is not equivalent in every rotation. In other words, whether the chlorine atom is next to the Bromine or the Fluorine does make a difference. If we call the F position 1, and the Br position 2, the Chlorine can then occupy 3,4,5 or 6. If the Chlorine is at the 2 position instead, the Bromine can then occupy 3,4,5 or 6. And there are many other possibilities to be added: we could have multiple halogen atoms, up to a maximum of six, in any combination of four elements: Fluorine, Chlorine, Bromine and Iodine. We can also use other radical groups: hydroxy, methyl, methoxy and ethyl groups being some of the most common, but there are other more exotic choices available. And also, the examples so far are just shown with one basic structure: the benzene ring. But the benzene ring itself can be changed, for example by substituting one of its carbon atoms with a nitrogen atom we get a pyridine molecule, and however many combinations of benzene derivatives we have gotten to, we can then double them by making pyridine derivatives of all of them. And if we are to substitute two atoms in a benzene ring with nitrogen atoms, we then have three different basic structures to copy everything on to. Not all of these structures are chemically viable, and because many structures are isometric, many are identical, but in general, whenever you have x different possible configurations of a molecule, adding another substitution to it provides x*(x+1) different new combinations. That is why you can go to a chemical catalog like this and find hundreds of chemicals in one specific category. Even with a basic structure and a few substitutions, the possibility of factorial mathematics lead to more chemicals than can actually be created and described.

This all might seem a little esoteric. After all, you (hopefully) don't have any halogenated benzene rings lying around your house. But the slight variations on organic molecules made possible by substitution are a gigantic part of our life. Estrogen and testosterone are the same basic molecule with some small substitutions of radical groups. Morphine and codeine are the same complicated molecule with one small difference: a hydroxy group changed into a methoxy group in one location. The difference between phenylephrine, a weak decongestant, and methamphetamine, is the removal of two hydroxy groups and the addition of a methyl group. A large part of what the pharmaceutical industry does is take the same basic structures and find slight variations on them, in the hope that they can find a drug that minimizes bad effects while maximizing good effects.

And because there are so many possibilities, so many variations on a theme, it is sometimes impossible to find what we are looking for. If we take a more complicated organic molecule, there may be literally ten million different variations on it. One of these will cure cancer. The other 9,999,999 will do nothing, or will be toxic. And there is often no way to know which one is the magic one. There is so much in the vast field of organic chemistry that is floating just outside of our grasp.

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