# Combinatoric Is Not Intuitive

Combinatorics is the study of relationships among several variables and how they act together. The number of combinations changes in ways our minds don’t find comfortable.

### An example of “Enumerative Combinatorics”

Suppose you have 24 people for a meeting and you want to seat them 6 to a table. How many ways are there to select those at the first table. the answer is 24 choose 6 and Google would give you the answer 134,596. It would take a while to consider all of them and choose the best arrangement. Of course, you would then need to assess 18 choose 6 for the second table, a mere 18,564, 12 choose 6 – a trivial 924 ways for table 3, and thank heavens just one way for the fourth table.

Obviously there are problems that we intuit an answer rather than solve it with precision. Most problems involving people solve intuitively

We use heuristics and trends, and guessing most of the time. The one thing we always use is the elimination of bad solutions. Maybe Jim and Joe must never be seated at the same table.

It pays to know you can get good answers without assessing all the choices.

### Some things are too complicated

Suppose I want to build a functioning model of the human brain. I start by knowing an adult human brain has about 86 billion neurons. In the neocortex, each neuron could have up to about 15,000 synapses connecting it to other neurons. So there are potentially 1.3 quadrillion synapses divided by two because they work in two directions. so about 650 trillion at the maximum.

That pretty much guarantees I’m not building a human brain. Even if I know which connections matter.

### Simplify

Fifteen thousand connections per neuron is a maximum. The average or median number might be very much lower. Some suggest 1,000 connections is average. Much easier to build. Now there are only 43 trillion connections. A lot, but better, but I have to arrange the connections.

Try to understand how many ways there are that could work. Combinatorics will help you understand how many ways there are to arrange the synapses. 3.6 sextillion. 86 billion choose 2 arrangements. A number with 21 zeroes. So I am not building that either.

### What it means

No single robot will be as capable at everything as any human. Each will be a specialist. Better than any human at what it is built for, but like humans outside its expertise, just dumb.

This has happened in the past with computers. Single purpose computers were no so rare when memory and processors were expensive.

Robots won’t take over. How many of them would you need, working in concert, to be vastly better than even one human? How would they co-ordinate themselves? How would they compete with humans acting in concert?

You can have some idea about what CAN happen if you understand the numbers.

### Have some fun

If you query Google with a search in the form “A choose B”, you can get answers that will astound you. how many ways are there for 5 people to pair. 5 choose 2, right. = 10. If you double 5 to 10, do you double the number of connections? No, 10 choose 2 is 45.

How about lottery possibilities? In Canada a national lottery is Lotto Max. where you choose 7 numbers from 50. 50 choose 7 is 99,884,400. Not much chance. Non zero, but unless your calculator has the ability to display more than 7 decimals the odds of winning will  show as 0. Close enough.

Lotto 6/49 is another lottery here.  you can choose 6 of 49 numbers in 13,983,816 ways. One more number and one more pick makes a big difference.

### Something to think about

How many relationships can you manage? In theory 150. Look up Dunbar’s Number.

If you know 500 people, how many ways are there to choose 150 that you want to build and maintain?  500 choose 150 is a huge number. You could look it up with Google. The answer has 131 zeroes. The number of atoms in the Earth has only 45 zeroes.

Want to be shocked? if you choose 150 out of 156 possibilities there are 18 billion ways to do it. Your maximum relationship list is random rather than carefully chosen. How many Facebook friends do you have?

### The meaning

Numbers don’t always tell the exact answer, but they can give you direction on how to think about it. Learn a tiny bit about combinatorics. Thank Google for the selection count option. the calculation would take a while with a pencil.

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