The reason for the sequence of questions is because there are two mathematical facts sitting right in the middle of this conversation:

• The Fundamental Counting Principle requires independent events to use.
• Both permutations and combinations involve dependent events due to a condition in the experiment that we cannot replace what we select. Therefore, there are always fewer options when you repeat the experiment.

So that brings us to a dilemma that we need to resolve: "How is it possible to use the Fundamental Counting Principle (which requires independent events) within something like the Permutation Formula (which requires dependent events)?" Let's drill down a little, using a specific example: 5 P 3 = (5)(4)(3) = 60. If this is an application of the Fundamental Counting Principle, then those must be independent events. The '5' is never a problem since the first selection is always independent of everything that came before it (which is nothing!). How can we describe the '4' and the '3' in such a way that they also represent independent events?

I Let Understand this by a problem ic., we have to fill 3 Vaccant seats from 5 person 5 ! So, Number of Ways canbes P3 = __ - : 5 x4 x3 = 60 (5-3)! Lp Here, box identies as Vacant 5 X 4 X 3 seat This seat can be only 3 person available to fill this filled by 5 persons vacant Seat i . c., P, , P2, P3, Py $ is only 4 Person available to fill this vacant Seat LD At each step we are selecting from vremaining option . So while events are dependent the pequential nature of the book - ess makes each step effectively independent within that Sequence i.c