Permutation question with a dial

[zaddyzad]

Homework Statement

So, you have a dial with 12 numbers (1 through 12), and you're wondering how many ways can you connect an number to another. So its therefor asking how many lines can you make with 1 - 12 in a circle.

Homework Equations

The Attempt at a Solution

I got the solution, I made 6 pairs one number to the one 3 numbers away from it (in a circular manner) and I knew if I were to offset everything by one id next a combination, and I could do this 11 times.

6 pairs * 11 offsets = 66 possible lines.


But I was wondering how to do this in a more mathematical manner (with factorials.) And I watched this interresting video on the enigma code "http://www.youtube.com/watch?v=G2_Q9FoD-oQ".

So I tried the same logic on the permutation the man used for the 26 letters. I attempted
12!/(6!^2 * 2^6) and didn't get 66.

How would you solve this using the logic that video used on the Engima code. I would simply like to know other ways to solve this equation

 

[haruspex]

You want the number of ways of partitioning 12 things into pairs? Pick one item at random. How many ways of choosing its pair? Now pick one of the remaining 10. How many ways to choose its pair? ...