I’m trying to understand concepts of probability from very basic principles. The first and the fundamental principle is the counting principle.
The counting principle: Stated simply, it is the idea that if there are a ways of doing something(Event A) and b ways of doing another thing(Event B), then there are a · b ways of performing both actions(Compound Event AB).
So the way I see it, every concept of combinatorics is derived from it.
For example: What are the number of ways to arrange ‘n’ objects in ‘k’ slots (n > k)?
Ans: There are ‘n’ ways to fill the first slot(Event 1), ‘n-1’ ways to fill the second(Event 2)…, ‘n-k’ ways to fill the kth slot(Event k). So by the counting principle there are nn-1…n-k ways to arrange ‘n’ objects in ‘k’ slots(Compound Event 1234…k).
Similarly how to arrive at nCk for ‘selection of k objects from n objects’ using the counting principle.
In other words how will you break the event of selecting k objects from n objects to smaller events so that we can use the counting principle to arrive at the expression nCk.
PS. I understand that we can arrive at the expression by looking at the sample space or by induction. I’m looking for deductions using the counting principle.