Permutation and combination

Two fundamental questions appear:

1. Does the order matter?
2. How many different selections are possible?

Permutation and Combination exist purely to answer these two questions.

Suppose you have 3 letters: A, B, C
You want to pick 2.

List all possibilities:

AB  
BA  
AC  
CA  
BC  
CB

Now ask:

• Are AB and BA different?
  If yes → order matters
  If no → order doesn’t matter

This single decision creates two entire systems.

Permutation (Order matters)

order matters: Different orders of the same items count separately.

Definition from logic:
You are choosing items AND their positions.

Example:
Choose 2 out of A, B, C where order matters.

AB ≠ BA
So both are counted.

Why the formula works

For n objects, choosing r:

For first position: n choices
For second: n-1 choices
For third: n-2 choices

So total ways:

n × (n-1) × (n-2) ... until r terms

This leads to:

Permutation = n! / (n - r)!

Combination (Order does NOT matter)

A name itself telling that what are combination so we dont care about order.

Same objects: A, B, C
Choose 2, but now AB = BA

So unique selections are:
AB
AC
BC

Only 3.

Here you only care about which items, not their sequence.

How combination is born from permutation

Notice:
Each unique pair like {A,B} creates 2 permutations:
AB, BA

So permutations overcount when order doesn’t matter.

To fix this:
Divide permutations by the number of internal arrangements of r objects:

Number of internal orders = r!

So:

Combination = Permutation / r!
            = n! / (r!(n - r)!)

Permutation counts arrangements
Combination counts groups