Suppose you have 10 tenis players competing on your tournament, and you want to figure out in how many different pairs you can organize them so that they can play their first match.
One way to calculate this is the following:
First pick one player at random. Now to pick his opponent you have 9 choices. After that pick another random player for the second match, and now to find his opponent you have 7 choices. If you keep repeating this the total number of pairs possible will be:
9 * 7 * 5 * 3 * 1 = 945
This process is called making a partition of 2 elements out of set A.
Now we can generalize this by saying that set A has 2n elements, so in the above example n = 5.
The total number of partitions we can make will then be:
(2n – 1) * (2n -2) * … * 5 * 3 * 1
And that is equal to:
(2n)! / (n! / * 2^n)