Yesterday I was hanging out with a friend and we ended up talking about combinatorics.

Eventually he asked me to work out the formula to find all the permutations of \(n\) unique objects. Perhaps this is a reminiscence of the combinatorics class I had in the bachelor and perhaps this is the way it is taught, but what came to my mind is a neat explanation:

If I have \(n\) unique objects, I can imagine having \(n\) spots. In the first spot, I can have any of the \(n\) elements. In the second spot, I can only pick one of the \(n - 1\) remaining elements and so on till I finish the spots and the elements.

This easily works out the formula:

\[\prod_{i=0}^{n-1} (n - i) = n!\]