Counting question- order irrelevance I have a super simple question about counting.
Imagine you have 4 objects. You can count them in whichever way you want (to be precise, $ 
_{4}P_{4}$ ways).  The well known property of order irrelevance applies-i.e. no matter how you count, you always arrive at 4.
My question is, is there any situation (measurement for instance), wherein switching the order in which orders are placed and/or counted changes the total count? If not, why not?
Many thanks!
 A: If I understand correctly, the question is, can the order in which we count things influence the count. I am not sure how to rigorously answer this but I'll try to give my two cents on what might be the appropriate intuition.
We can define the number of objects in a (finite) collection $C$ as the smallest $n$ for which there exists a bijection $f : \{1,\ldots, n\} \to C$. Then we 'count' them for instance as $f(1),f(2),\ldots, f(n)$. If ordering played a role, then there would be another bijection $g:[n'] \to C$ with $n\neq n'$, but this cannot happen as then there would be an isomorphism $\{1,\ldots, n\}$ to $\{1,\ldots, n'\}$.
A real world example that I can think of where the order may play a role: Suppose you are counting a collection of animals. Perhaps if you order them so that the lion is next to the lamb, and far down the line, then by the time you arrive at the lion, you find the lamb missing.
Another example may be considering a collection of radioactive isotopes, and seeing how many have not decayed. In this case even the speed at which we count is important.
The moral behind both cases is I think that real world counting may be flawed. We are interested in the instantaneous count of some objects, that is, how many are there right now. In the examples, the method of counting is  flawed to such a degree that the sheer act of carrying out the counting may influence the result.
