# How to calculate the number of Combinations (cPr) of a Permutation (nPr)

Im working on a problem where i have 2 variables

A= (1,2,3) B= (3,2,1)

I have to calculate the number of permutations, and the number of paths by swaping adjancent numbers until A becomes B

For permutations Im currently using the following function in python

nPr = (math.factorial(n))/(math.factorial(n-n))


This one works as intended and the result is ok (in this case nPr = 6), however how can I calculate the number of paths (swaps of pairs) until i get to B?

The combination formula

(n!)/((n-r!)*(r!))

doesnt seem to make the job since it always comes out as 1. (when r = n)

So there has to be another way to do this, I tried using the factorial of nPr however when 6! = 720, meaning theres 720 posible paths to get to B and thats not true.

this was just one example of 3 digits but A and B can be sets of numbers of 2,3,4,5... etc

• Welcome Math.SE! Take the tour to get familiar with this site. Mathematical expressions and equations can be formatted using LATEX syntax. If you receive useful answers, consider accepting one. – Shailesh Sep 28 '15 at 4:17

## 1 Answer

For a list of size n my idea would be:

You need maximum n-1 swaps to put the first digit into its correct position, then you need maximum n-2 swaps to put the second digit into its correct position, n-3 for the third and so on untill 0 for the last.

Therefore maximum (1/2)(n-1)(n) swaps