Intuition behind permutations and combinations (2016)

I remember reading this excellent little book with very easy to understand explanations of some of the basics re. permutations and combinations, back in school:

"Mathematics of Choice: How to Count Without Counting" by Ivan Morton Niven

https://www.amazon.com/Mathematics-Choice-New-Mathematical-L...

Why do we need this? What's wrong with good old "there are n ways to pick first element, n-1 ways to pick second, etc and then apply burnside to glue together things that we treat as identical"?

Wow. very different and useful mental model to think combinations and permutations.