What are the chances of that?

In the book I'm writing at the moment I'm considering the relationship of the arrow of time to entropy, the measure of the disorder in a system that comes out of the second law of thermodynamics. Entropy can be calculated by looking at the number of different ways to arrange the components that make up a system. The more ways there are to arrange them, the greater the entropy.

As an example of why this is the case, I was talking about the letters that go together to make up that book, and the very specific arrangement of them required to be that actual book. Assuming that there will be about 500,000 characters including spaces in the book by the time it's finished, then there are 500,000! ways of arranging those characters. That's 500,000 factorial, which is 500,000x499,999x499,998x499,997... - rather a big number.

It's not practical to calculate the number exactly, but there are approximation techniques, and if the large factorial online calculator I found is correct, then 500,000! is around  1.022801584 x 102632341 - or to put it another way, around 1 with 2,632,341 zeros after it. That's a big number. By comparison there is just one way to arrange the letters to make my book*. So by producing the book I have vastly reduced the entropy. This seems to run counter to the second law of thermodynamics, which says that entropy in a closed system should stay the same or increase. But the clue is that get out clause, 'closed system'. The book isn't a closed system - the arrangement of the letters has come out of my head as a result of the consumption by my brain of a fair amount of energy. And it's that energy that makes the reduction in entropy possible.

Good stuff, but it shows that we shouldn't expect a room full of monkeys to come up with the complete works of Shakespeare - or my book - any time soon.

* This is only true if you consider each letter 'a' to be different from each other letter 'a' - imagine, for instance, each letter has a serial number. In that case it is literally true. In reality I could make what appears to be exactly the same book but swap all the letter 'a's with other letter 'a's and it would read exactly the same. And of course the same applies to every other letter. But there are still vastly fewer ways to organize those letters to spell out the same book than there are of producing any pattern whatsoever.