# In combinatorics, how can one verify that one has counted correctly?

This is a soft question, but I’ve tried to be specific about my concerns. When studying basic combinatorics, I was struck by the fact that it seems hard to verify if one has counted correctly.

It’s easiest to explain with an example, so I’ll give one (it’s fun!) and then pose my questions at the bottom. In this example there are two ways to count the number of ways $2n$ people can be placed into $n$ partnerships (from the book Introduction to Probability by Blitzstein and Hwang):

Story proof for the right side: For the first person you have $(2n - 1)$ choices for partner. Then for the next person you have $(2n - 3)$ choices, etc.

Story proof for the left side: Line all $2n$ people up, walk down the line, and select every 2 people as a pair. The ordering you chose for the line determines the pairs, and there are $2n!$ ways to order $2n$ people. But divide by $2^n$ because the order within each pairing doesn’t matter. Also divide by $n!$ because the order of the pairings doesn’t matter.

My question is:

What if I had been tasked with getting this number at work, and I chose the approach on the left side, but I neglected to divide by $2^n$ and only divided by $n!$? I’d be wrong of course, but how could I have known?

I suppose one answer to my question could just be “Try hard, try to think of ways you might be over/undercounting, look at other examples, and continue to study theory.”

But the prospect of not knowing if I’m wrong makes me nervous. So I’m looking for more concrete things I can do. So, three questions of increasing rigor:

• What are specific “habits of mind” that people in combinatorics use to avoid error?
• What are specific validation techniques that such people use to check their work? (One does come to mind from my example: Calculate the same thing two ways and confirm equality)
• Is there any formal list of questions that, if answered, should verify that one’s approach is correct?